Savvas Mousionis
email: mousioniz@hotmail.com

Alex Vakaloudis
email: avakaloudis@hotmail.com

Constantinos Hilas
email: chilas@teiser.gr

A Study on the Security, the Performance and the Penetration of Wi-Fi Networks in a Greek Urban Area

Keywords: Wi-Fi networks usage, wireless security, war driving, urban networks

This paper presents a study on the expansion of urban Wi-Fi networks and the degree of users' awareness about their characteristics. It involves an experiment contacted at the area of Serres, a Greek city of around 70,000 inhabitants. The findings revealed that although the number of Wi-Fi networks is quite high, their owners are unaware of their technical settings. As a result many networks remain either unlocked or with WEP encryption while many adjacent networks use the same channel thus reducing their performance.

Introduction

It has been around seven years since the introduction of Asymmetric Digital Subscriber Line (ADSL) as a high speed Internet access service in Greece. Despite the initial reluctance by home users to upgrade their previous dial-up connections to ADSL, currently a large number of Greek homes or enterprises connect to the internet through ADSL. It is a mainstream practice for providers to supply a wireless modem/router/access point for every new connection, mainly in the form of a subscription gift. This has gradually filled Greek cities with Wi-Fi networks. These devices typically use the 802.11b/g protocols with a 100mW antenna at 2.4 GHz. Transmission occurs in one of 13 overlapping channels [1].

The present paper explores the use of the resulted Wi-Fi networks which in turn provides clues for the degree of user awareness on wireless security. The main tool used in this research is the method of war driving [START_REF]Wi-Fi Wardriving and Warchalking[END_REF]. War Driving is the act of searching for Wi-Fi wireless networks by a person in a moving vehicle using a Wi-Fi equipped computer, such as a laptop or a PDA. It is similar to using a radio scanner, or to the ham radio practice of DXing.

War driving is a play of words on the older term war dialing, which is automatically calling various telephone numbers to look for any that have a modem attached. War dialing, in turn, comes from the 1983 movie War Games now written in the cult lore of computer geek circles. In the movie a young cracker (Matthew Broderick) is using war dialing to look for games and bulletin board systems. However, he inadvertently ends up with a direct connection to a highlevel military computer that gives him control over the U.S. nuclear arsenal [START_REF] Brown | WarGames: A Look Back at the Film That Turned Geeks and Phreaks Into Stars[END_REF].

The paper proceeds as follows. The next section defines the hypotheses of the experiments, its subjects, methods and materials and the problems that occurred during its execution. Section 3 lists the results, while in the last section the outcomes of our work along with suggestions for improvements and future work are discussed.

Description of the Experiment

Motivation and Hypotheses

The survey discussed in this paper has two objectives. The first one is to investigate the penetration of Wi-Fi networks considering the provincial city of Serres as a case study. The outcome may not only be used to demonstrate their wide spread but also to confront the reluctance of home users to operate a wireless network due to health considerations. Measurements of the Signal-to-Noise Ratio (SNR) will confirm that the whole city is covered by a number of wireless networks. Therefore, if someone is surrounded by neighbors owning Wi-Fis, she is already exposed to some RF radiation anyway. Moreover, this RF exposure is normally thousands of times below international standards [START_REF]World Health Organization: Electromagnetic fields and public health, Base stations and wireless technologies[END_REF].

Our second objective is to examine the manner of usage of Wi-Fi networks and to demonstrate the ignorance of their owners when it comes to simple security of performance settings. The knowledge, for example, to choose the type of encryption or to adjust the Wi-Fi channel is important for the good operation of a network, yet the vast majority of wireless access point (WAP) users are either oblivious or inconsiderate when it comes to their network.

Although WiFi technology security vulnerabilities are well known, the extent of these vulnerabilities may be surprising. War driving may identify many potential points of entry [START_REF] Berghel | Wireless infidelity I: war driving[END_REF].

Subjects

Serres is the capital of the Serres Prefecture located in the Central Macedonia Periphery of Greece. The city has a population of around 70,000 inhabitants (56,145 in the last official census of 2001) and is an important trade centre for tobacco, grain, and livestock. In our view, it represents a sound choice for an experimental subject since it reflects the average situation regarding wireless networks in Greece. It is situated in a location which is neither near the cuttingedge capital (Athens) nor one found in less developed places. For the better interpretation of results, we divided the city into the following areas:

-City centre: The area where mainly commercial shops or companies exist.

Wireless networks in this area are expected to have been setup by professional technicians.

-Around the city centre: A residential area with few shops or companies.

-Suburban and densely populated: A residential area with block of flats of four to six floors. -Suburban and sparsely populated: The outskirts of the city, a residential area with houses. -Student area: The area around the technical university (T.E.I. of Serres) populated with students. Some of them study in the Informatics and Communications Department, hence a higher degree of technical expertise and involvement is expected from them. -Difficult to approach: The area consisting of the hills around the old part of the city with old houses and narrow roads.

Methodology and Tools

The method used to carry out the experiment is war driving, in other words, driving around the city and stopping regularly to discover wireless networks. The density of these stops is specified by:

-The surrounding area: In densely populated areas stops are every 10-15m since networks of the top floors must be discovered and the number of expected networks is high. On the other hand, in sparsely populated areas the scheduled stops are around 20-30m. -The surrounding environment: Parked coaches and large trees hinder the discovery of networks.

The tool used is Network Stumbler which is a widely used tool that provides all required information [6]. This is the WAPs MAC address and SSID which were used to identify unique networks, the communication channel, the encryption standard, the type of the device, e.g. WAP or station, and the SNR.

Regarding the first objective of our experiment we measure:

-The number of unique Wi-Fi networks at each stop. The uniqueness requires checking if a network has appeared in adjacent measurements. -The SNR of each network and the maximum SNR for each point of measurement.

For the use of Wi-Fi networks we examine the following parameters:

-The encryption used, i.e. no encryption, WEP, WPA and WPA2. In the case of unencrypted networks we also try to login to the administration console to check if the user has changed the default username and password -The SSID used, i.e. if it is the default one or it has been altered -The Wi-Fi channel to which the network is adjusted.

From a legal perspective, there is no restriction of examining wireless networks broadcasting in a public place, especially for academic purposes. Moreover, we have just searched for the existence of wireless networks in a non-intrusive way, with no ulterior motive. Adding to this, no attempt was made to interfere or jam the wireless traffic nor did we try to correlate the networks or their traffic with specific persons. We, also, do not publicize the exact location and owner of the individual insecure APs. What is illegal is the unauthorized access to a wireless network in order to steal internet access, steal information, alter the network's configuration or commit other computer crimes. As a result, for networks that were found unlocked, our action stopped to the point of examining whether the user still uses the default security settings. We want to stress out that this may not always be legal in other countries.

Problems

Although war driving seems to be a time-consuming, yet straightforward process, a number of issues appeared during its execution. Stopping even for one minute in the city centre is not always permitted or it may cause the annoyance of other drivers. Consequently, it may take a few attempts to take a single measurement. Likewise, a stop on a street with parked cars on each side causes interruption of the traffic. Furthermore, some streets are one-way traffic which increases the time needed to reach a desired point. Finally in the difficult to approach section, driving must be very cautious to avoid damaging the car. There were cases where war driving turned out to be war walking.

Since the experiment took place during the summer, the leaves of the trees were a source of reduced SNR values. Hence, the measurement should not take place at points below or nearby large trees.

Finally, an updated map had to be used since the city is expanding and an outdated map was inconsistent with the real picture; for instance there were sections of roads that had been widened or even replaced by squares or roundabouts.

All the above, while solvable, increased the expected time planned for each session of war driving.

Results

The war driving part of the experiment took place over the three summer months of 2010. It took 14 sessions of 10 hours each to cover the whole city. The processing of the results included printing the screenshots of Network Stumbler, identifying unique networks along with their maximum SNR.

Overall, 1021 measurements were made and 5374 Wi-Fi networks were found (Table 1). From the 677 (12.6%) unlocked ones, 268 had retained the default access to the administration console. As regards the use of encryption 840 (15.6%) networks were using WEP, 3782 WPA and just 75 WPA2. 3728 were still using the default SSID and the rest 1646 had changed their SSID. The average SNR for each measurement point was 34.2 dB. Considering that WEP encryption is only little better than no encryption we see that almost one third of the networks are susceptible to eavesdropping.

The channels used for Wi-Fi are separated by 5 MHz in most cases but have a bandwidth of 22 MHz. As a result channels overlap and it is possible to find a maximum of three non-overlapping channels. Therefore, if there are adjacent pieces of WLAN equipment that need to work on non-interfering channels, there is only a possibility of three. In Table 2 the channel usage of the surveyed networks is shown. Channels 1, 6 and 11 do not overlap with each other and are the preferred choices when setting up a WAP. Channel 6 is usually the default factory setting. The findings in Table 2 coincide with the common practice. Interestingly, there are WAPs set up to work on channels other than the three most common. In Fig. 1 the position of the discovered networks in the city is depicted. The number of identified networks at each location is visualized by means of spheres of variable sizes. The size of each sphere correlates to the number of networks at the point. In Fig. 2 the SNR measurements at each position are visualized. One may observe that the city seems covered with Wi-Fi signals. In Fig. 3 the findings of Table 2 are further clarified. The bar chart displays the possibility two or more adjacent networks to use the same Wi-Fi channel.

Conclusions and Discussion

Interpretation of Results

This experiment demonstrated that a medium-sized city (for Greek standards of population) is almost fully covered by Wi-Fi networks. The only blank points where squares or the outer limits of the city. The number of unique networks discovered is translated to one network per around 11 inhabitants. This indicates a good penetration of the wireless technology plus that there is still room for more networks to be added. It also illustrates a wide-spread interest of people in Greece for wireless internet access.

As the number of base stations and local wireless networks increases, so does the RF exposure of the population. Recent surveys have shown that the RF exposures from base stations range from 0.002% to 2% of the levels of international exposure guidelines, depending on a variety of factors such as the proximity to the antenna and the surrounding environment. This is lower or comparable to RF exposures from radio or television broadcast transmitters [START_REF]World Health Organization: Electromagnetic fields and public health, Base stations and wireless technologies[END_REF].

Our measurements of the SNR also confront the fear of not installing a home wireless network because of health risk concerns [START_REF] Bale | Health fears lead schools to dismantle wireless networks[END_REF]. If a specific apartment is surrounded by Wi-Fi networks, it is already susceptible to their electromagnetic radiation. However, considering the very low exposure levels and research results collected to date, there is no convincing scientific evidence that the weak RF signals from wireless networks cause adverse health effects [START_REF]World Health Organization: Electromagnetic fields and public health, Base stations and wireless technologies[END_REF].

An important question, left to be answered, is whether users actually access the Internet over a wireless and not over a wired medium. In other words, we can not be sure whether all the discovered networks serve their purpose or are idle.

On the issue of configuring a Wi-Fi network, our evaluation of the experiment's results is carried out under the perspective that the performance and security of wireless networks are topics of general interest, regularly brought up in ordinary discussions. Users are concerned with security and desire faster internet access.

At the same time, the basic configuration of a wireless access point is a procedure that does not require advanced technical expertise or understanding of the wireless technology. In addition, most devices are shipped, by the ISPs, with manuals detailing simple settings alterations e.g. how to change the channel or to apply encryption. And while encryption or Wi-Fi channel numbers are notions whose exact meanings are unknown to the common user, the provided web-based interfaces are simple enough to facilitate their management.

Nevertheless, our survey revealed that despite the importance of security and the easiness for its application, very few users are confident enough to change their WAP settings. As a result, only 1.4% encrypts their data with WPA2. The only exception to this rule is the city centre where 7% of the networks use WPA2. Even though this can be explained by the higher number of business related networks, these are usually setup by specialized technicians and hence this percentage should have been much higher. No WAP using WPA2 was found in the student populated area which took us by surprise as we expected to find more technically competent users there.

It was found that around 70% of Wi-Fi networks are encrypted with WPA. We assume that this is because WPA is the default encryption for the devices preset by Greek ISPs in the last couple of years. Older subscribers still use the unsecured WEP or no encryption at all.

Another evidence of security unaware users is the number of default SSIDs found. Keeping the default SSID can not be considered as a security risk on its own, but it prompts a potential hacker to try to penetrate to such a network. Default SSIDs provide clues about the apparatus model and imply that there is a good possibility that the default administration authentication has also been kept.

The ignorance of users is further exposed when it comes to channel collisions, i.e. the use of the same Wi-Fi channel by many neighboring networks. It is found that when interference exists, the throughput of the system is reduced. It therefore pays to reduce the levels of interference to improve the overall performance of the WLAN equipment. Although users crave for faster internet access they do not take the corresponding actions. Our survey shows that in 87% of the cases another nearby network was using the same channel. This causes interference among such networks and reduces their throughput.

Relation to Other Works

To our knowledge no similar experiment has been contacted in Greece. A study on war driving in Dartmouth college campus is published in [START_REF] Minkyong | Risks of Using AP Locations Discovered Through War Driving[END_REF], while the Professional Information Security Association (PISA) of Hong Kong reports the findings of war driving in Hong Kong and Macau [START_REF] Fong | PISA & WTIA's Hong Kong & Macau War-Driving Report[END_REF].

The first one is a survey that focus primarily on the accuracy of the WAP position estimation and the impact it has on pervasive-computing applications that depend on knowledge of user location. The article also comments on the effect of using estimated WAP locations in computing AP coverage range and estimating interference among WAPs.

The findings of the second survey are similar to ours, although it was made 3 years earlier (2007). An increasing adoption of encryption settings was identified although 72% of the encrypted sites used WEP. Also, more that 40% of the WAPs kept the default SSID settings while 20% of the rest used individual/family names or organization names as their SSID.

Also, RSA, the security division of EMC 2 Corporation, has commissioned annual research over the past seven years, as part of its campaign to promote and improve best practices in wireless security [START_REF]RSA wireless security surveys[END_REF].

Impact for Practitioners

The fact that Wi-FI APs come with WPA encryption on their default settings has improved the overall security of home networks. We propose a similar approach to be adopted regarding the Wi-Fi channel setting. Randomizing the assigned channel per WAP will reduce the probability of interference and thus will improve performance.

It should be noticed that although all manufacturers provide advanced security measures in their appliances such as modifiable network identifier names and passwords, address filtering, firewalls and WPA to protect wireless networks, it is the consumer who must make the final steps in order to install, configure and adjust all features for maximum security. Thus, it would be very helpful to spare a few pages in user manuals with detailed step-by-step guides on security and performance.

The ignorance or fear to manipulate device settings seems to be apparent in the behavior of professional technicians as well and this is a situation that has to be addressed.

Research Agenda

The same experiment will be repeated next year to examine changes in the penetration of networks and the use of channels/encryption so as to find out any progress in these issues. We also plan to perform similar experiments in other cities in order to compare the relation of the public to wireless networks in different areas. The parameter of checking MAC filtering will also be added.

Fig. 1 .

 1 Fig. 1. Positioning and population of Wi-Fi networks. The size of the spheres correlates to the number of networks.

Fig. 2 .

 2 Fig. 2. Wi-Fi coverage of the city. Measured SNR values are visualized by the size of the sphere.

Fig. 3 .

 3 Fig.3. Percentage (per channel) of neighboring networks that were found using the same channel.

Table 1 .

 1 Number of Wi-Fi networks found at each of the areas described in subsection 2.3

	Area	Unlocked WEP WPA WPA2 Default Altered Total
					SSID SSID
	City Centre	103	68 521	52	459	285 744
	Around City Centre	181 276 1186	13	1199	457 1656
	Densely populated	167 235 820	10	867	365 1232
	Sparsely populated	64	63 217	0	258	86 344
	Student Area	41	81 211	0	1199	457 1656
	Difficult to approach	121 117 827	117	730	335 1065
	Total	677 840 3782	75	3728	1646 5374

Table 2 .

 2 Active networks per channel

	1	2	3	4	5	6	7	8	9	10	11	12	13
	1575	40	34	38	45	2296	50	45	57	42	1107	12	33


Bidisha Ghosh
email: bghosh@tcd.ie

Vikram Pakrashi
email: pakrashv@tcd.ie

Franck Schoefs
email: franck.schoefs@univ-nantes.fr

High Dynamic Range Image Processing for Structural Health Monitoring

Keywords: High Dynamic Range (HDR) Imaging, Structural Health Monitoring (SHM), Pitting Corrosion, Damage Detection, Image Processing

This paper proposes High Dynamic Range (HDR) imaging as a protocol for structural health monitoring for the first time. HDR imaging protocol has been experimentally validated on images of pitting corrosion in this paper and has been further applied to isolate a range of image backgrounds that arise out of a number of environmental conditions. The superiority of HDR imaging over a standard imaging process has been demonstrated. The results indicate that the proposed protocol is not limited by the examples considered in this paper and can readily be applied to a number of infrastructure maintenance management related applications. The protocol complements and improves standard visual inspection techniques and expert opinions. This approach immediately lends itself to further mathematical and statistical analyses, both qualitative and quantitative.

Introduction

A four step process is usually popular in the field of Structural Health Monitoring (SHM) and damage detection (Rytter 93). The first three steps are connected and involve the detection of the presence, the location and the severity of damage, while the fourth step, related to the prediction of service life is usually a separate problem by itself. In most of the world infrastructure systems or parts of the systems are continuously ageing and deteriorating. This in conjunction with limited government and private funding emphasizes the importance of infrastructure maintenance management and prioritisation through SHM (Žnidarič et al., 2005) Intrusive, non-intrusive and semiintrusive testing are often carried out following visual inspections. The test results can be readily fed into Bridge Management Systems (BMS) or Infrastructure Management Systems (IMS). This approach is not specific to particular management processes for infrastructure networks and can be used to significantly affect decision making procedures and financing for infrastructure owners, managers and end users. Decision making based on testing even accommodates a more modern infrastructure specific probabilistic hierarchical approach [START_REF] Pakrashi | Reliability based assessment of structures in marine environment[END_REF].

Among the various methods of testing, nonintrusive or Non Destructive Testing (NDT) affects the tested structure to a minimum level. The quality of results and the cost efficiency of such methods are extremely important as they directly affect the performance indices of a structure or a network of structures in terms of safety and durability at a given time. The methodology and the protocol of NDT directly relate to the quality and the quantity of transfer of benefits of an infrastructure system to the end user.

A typical and specific application in this regard can be found in the field of corroding infrastructure systems, in particular underwater marine structures undergoing pitting or any other form of corrosion. These structures often require underwater inspections and are carried out by trained divers who usually are not structural experts [START_REF] Rouhan | Probabilistic modelling of inspection results for offshore structures[END_REF]. Traditionally, photographic and video recordings are available from diving campaigns. Descriptive and expert ranking based classification of damages are standard outputs from these data. Even when more detailed surveys and experiments are carried out, only pointwise information is generally available [START_REF] Boukinda | Marine growth colonization process in Guinea gulf: Data analysis[END_REF]. Though a natural choice, the use of image processing in classifying damage conditions has not been used to the potential it holds and no testing protocol currently exists. Image processing based detection of pitting corrosion, although reported in literature [START_REF] Itzhak | Pitting corrosion evaluation by computer image processing[END_REF][START_REF] Choi | Morphological analysis and classification of types of surface corrosion damage by digital image processing[END_REF] , has not received significant attention and the quality of detection obtained as a function of image conditions has been barely looked into [START_REF] Pakrashi | ROC dependent event isolation method for image processing based assessment of corroded harbour structures[END_REF]. It is felt by the authors, that image processing based detection of corrosion can be significantly improved for a wide range of problems through the specification of certain protocols. Such protocols give rise to a number of opportunities well beyond standardising and homogenising test methods and the comparison of test results under varying environmental conditions.

The specification of an improved imaging protocol partially offsets the inexperience or the lack of expertise on the part of the diver and enhances the quality of input data to the analysing engineer, scientist or expert. The significant increase of the quality of the input data without the requirement of additional costs in terms of sensors or the training of individuals is extremely important. In addition, an image based approach can make a significant amount of archived image and video data available from previous inspections over a number of decades useful through correct specifications. Image processing based output data readily lends itself to mathematical and statistical treatments and the quantification of output results are significantly more correct, detailed and sophisticated. All of these advantages greatly improve the credibility of the assessment of a structure at a given time and the evolution of a structure over a period of time.

This paper proposes High Dynamic Range (HDR) imaging as a protocol for NDT based SHM. HDR imaging involves merging of a number of photographs of the same object taken at various different exposures [START_REF] Battiato | High dynamic range imaging for digital still camera: an overview[END_REF]. The method is quick, can be performed with a standard digital camera, requires no new sensor and demands little training or experience from the individual logging the data. HDR imaging is particularly useful for targets that are relatively static over the period of time during which the photograph is taken [START_REF] Jacobs | Automatic high dynamic range image generation for dynamic scenes[END_REF]. HDR has been successfully applied to other fields and applications, ranging from bioengineering (Wandell et al., 2002) to art (Colbert et al., 2005) and photography (Pell 09) to dramatically improve the quality of an image with relatively little extra effort.

To the authors' best knowledge, there has been no application of HDR imaging for SHM although it holds exceptional potential to do so. This paper applies the proposed HDR imaging protocol successfully on a number of images of steel piles belonging to on-pile wharves corroding in the spray zone of a marine environment. The involvement of expert opinions in such detections is discussed. The paper also investigates the identification and classifications of the image backgrounds. This adds a relatively new possibility to the standard four step SHM method where identification of the type of damage as well as the presence, the location and the severity is possible.

High Dynamic Range (HDR) Image Processing

The dynamic range of a real-world scene is the ratio between the radiance values of the brightest and the darkest parts of the scene, whereas for a display the dynamic range is the ratio between the maximum and the minimum intensities emitted from the displayed image [START_REF] Ward | High dynamic range imaging: Acquisition, display and image-based lighting[END_REF]. In real world every scene has a significant amount of brightness variation (Figure 1). The human eye is capable of handling a dynamic range of 100000:1. In simpler words, the human eye can accommodate the dynamic range between the shadowy interior of a room to the sunlit views outside the window. In contrast, a typical video camera or a digital still camera provides a dynamic range of 1000:1. Hence, photographers have to decide on a limited range of radiance values which can be captured within the limited dynamic range of the camera. Any object with radiance values above or below this camera-specific dynamic range will be captured as having the maximum or minimum limiting radiance values of the specified range respectively. As a result any real world scenes with shiny metals, sun, artificial light sources or shadowy details will pose a problem to the photographer. In these cases, the captured images will be either too dark in some areas or too bright in some. As a result, in image-processing based SHM we are often left with photographs of significantly compromised quality for interpretation and detection purposes. The low range of brightness information of these images poses a severe limitation on what computational vision can accomplish. A very simple modification, HDR imaging, can be made to any conventional imaging system to dramatically increase this range.

An HDR image is such an image whose dynamic range exceeds the dynamic range of the capturing or the display device and in turn provides more information on every pixel of the image. The HDR images with composite radiance map can be created by taking multiple photographs of the same scene with different exposures of using a standard digital camera [START_REF] Debevec | Recovering high dynamic range radiance maps from photographs[END_REF]. By using multiple exposure time each photograph in the sequence will have some pixels of the image properly exposed, and other pixels will be either too dark (underexposed) or too bright (overexposed). For example, a photograph captured with a long exposure time will be overexposed in the bright areas but will capture the details of the dark regions very well. A small exposure time will ensure that the details in the bright regions are captured well, but the photograph will be void of any details in the shadowy regions. The complementary nature of these images allows one to combine multiple photographs into a single, high dynamic range image whose pixel values are proportional to the true radiance values of the original scene.

High Dynamic Range (HDR) Image Processing

Of the many different types of damages that significantly affect structures, pitting corrosion is a specific problem where HDR imaging can be used successfully. This is a commercially important problem as it not only affects a very wide range of civil engineering structures, including piles [START_REF] Liang | Reliability analysis for the existing reinforced concrete pile corrosion of bridge substructure[END_REF], bridges [START_REF] Darmawan | Spatial timedependent reliability analysis of corroding pretensioned prestressed concrete bridge girders[END_REF] and pipes [START_REF] Shivaraj | Ultrasonic circumferential guided wave for pittingtype corrosion imaging at inaccessible pipe-support locations[END_REF] but also a range of fields as varied as from aviation [START_REF] Qingyuan | Evaluation of the probability distribution in pitting corrosion fatigue life in aircraft materials[END_REF] to the printing industry (Sailer 07). Pitting corrosion usually creates isolated corroded units within metal and is typically characterised by their rapid change in depth over a relatively short geometric spread. Pitting corrosion is also, very typically, associated with considerable change in colour with respect to the original metal background due to the chemical reactions that bring about the corrosion (Tsushima et al., 1997). Detection from the typical discolouration and contrast with the background surface is generally carried out using photographs. The HDR imaging protocol assimilates the ability of human eye to detect a broader dynamic range and enhances the capacity of detection.

For the purposes of illustration, we consider photographs of steel piles belonging to on-pile wharves corroding in the spray zone of a marine environment of the Atlantic Coast, Estuary of river Loire, France. In these photographs a wide dynamic range has to be accommodated as steel may have a shiny metallic surface. Pitting corrosion occurs on the steel surface over isolated regions and is often of contrasting colour with respect to the background. Three images of the same corroding area undergoing pitting corrosion are captured with a standard commercially available digital camera ('Bridge' digital camera, 9.1 Megapixels (maximum: 3456 x 2592 pixels, sensitivity ISO 80 to 3200, Shutter speed: 30 -1/4000 sec, opening: 2.8 -5.2, focal length (KB): 28 -560). One of these images is normally exposed, the second one is overexposed and the third one is underexposed. The amount of over and under exposure is set to one f-stop where the f-stop is defined as the ratio of the focal length of the lens to the diameter of the aperture of the lens. The three photographs are combined together to generate the HDR image following the algorithm developed by [START_REF] Debevec | Recovering high dynamic range radiance maps from photographs[END_REF]. The algorithm considers multiple images of the same scene with different exposure times. These multiple images are used to recover the response function of the digital camera used to capture the photographs. The response function maps the actual quantity of light impinging on each element of the sensor array in the camera to the pixel values that the camera outputs. With the recovered response function the algorithm fuses multiple input images to a single HDR radiance map whose pixel values are proportional to the true radiance values in the scene. Figure 2 shows the HDR image along with the three parent images. The use of a higher number of parent images, above and below the normal exposure can improve the final quality of the HDR image. However, this requirement is case-specific and for the present situation three images were found to be sufficient to demonstrate the effectiveness of an HDR image.

Figure 2. The generation of a High Dynamic Range (HDR) image.

A line is chosen over a section of the HDR image straddling both the corroded and the uncorroded regions and the variation of Red (R), Blue (B) and Green (G) coordinates of the pixel values on that line are observed and compared with the pixel values (RGB) of the same line drawn over a normally exposed image. This comparison is presented in Figure 3. The horizontal axis refers to the local pixel coordinates on the chosen straight line along its direction on the image. The vertical axis is the axis of RGB values ranging from 0 to 256. An RGB triplet (0,0,0) refers to a completely black pixel while (256,256,256) is a perfectly white pixel. In Figure 3, the corroded zone exists from pixel 150 to about 300. The normally exposed image consists of a significant number of points with low RGB values, especially within the corroded region. This is because the corroded zone is too dark and the radiance values are outside the camera-specific dynamic range. The relative darkness within the corroded zone strips the image with normal exposure of a significant amount of information or detail that could have been observed with naked eye given the exact same lighting conditions. On the other hand, it is clearly observed that the tonal range of the HDR image is wider than that of a normally exposed image. Additionally, in HDR, a general shift of the pixel values is observed towards the higher side, the colour variations within the corroded zones are better exposed and the difference between the corroded and the uncorroded zone is fully exploited. Similar exercises have been carried out on underexposed and overexposed photographs against the HDR image. On each occasion, the HDR image was observed to be of superior quality based on the information retained. These exercises are not separately presented to avoid repetitive results. It is observed in Figure 3 that the RGB values comprise of a trend and some relative high frequency fluctuations, quite akin to noise. Since the detection is dependent relatively more on the trend patterns than that of the fluctuations, a wavelet based denoising (Misiti et al., 2008) has been carried out on each of these pixel plots of RGB values. Figure 4 plots these pixel value trends for the normally exposed and the HDR image simultaneously. The increased tonal range and the relatively superior information content of the HDR image are quite apparent. Consequently, a wavelet based denoising or any kind of denoising of a similar nature is proposed as a post-processing for similar detection or identification processes. It should be noted, that Red, Blue and Green pixel values are linearly combined to produce various colours and the variation of tonal range will depend on their relative participation to produce a certain colour. This explains the relatively less variation of Red in Figure 4.

The variation of the RGB pixel values along a certain line on an HDR image straddling zones of contrast is further explored for corroded steel samples featuring marine growth and dusty condition respectively. Figure 5 shows the RGB plot over linear sections with and without marine growth. The sensitivity of the values to changing situations is apparent. Figure 6 shows an image with dusty conditions. This is a particularly interesting case where staining due to corrosion occurs and can be expected to be picked up through the difference of colour intensities. On the other hand, dusty pitting zones exist where the difference of colour intensities within the zone of pitting may not be the guiding factor for detection. The pitting zone can be detected at its edges since the depth of the corroded is represented as a sudden change in relative darkness or brightness with respect to the prevailing background. A linear section is chosen in Figure 6 straddling both the stained and the unstained part. The corresponding RGB values of the pixels are observed to have changed significantly when the line enters the stained zone at around pixel values 400 to 500. The RGB values of the pixels, where the chosen line straddles the corroded and the uncorroded zone with similar backgrounds are observed to be significantly changing pointwise at the entry and at the exit locations near to pixel values 100 and 200 respectively.

HDR Imaging Protocol for Image Background Characterisation

It is observed in the previous section, that the variation of the RGB triplet is sensitive to the type of surface that is being photographed. It is often quite an important problem to identify the kind of surface background that is present. Identification of a surface background can be traditionally and correctly carried out by experts through visual inspection. However, it would be of significant use if expert opinion on a range of visual observations can be used as an initial classifier to characterise various backgrounds automatically. If disparate image backgrounds of engineering interest readily form well-distinguishable and relatively nonoverlapping clusters while similar image backgrounds relatively coagulate within a three dimensional RGB space, such a classification will be deemed feasible and applicable to real problems. A somewhat similar approach has been applied before in the field of natural image processing [START_REF] Naccari | Natural scenes classification for color enhancement[END_REF].

For this purpose several realisations were carried out for a number of similar and dissimilar image backgrounds from the specimens and a three dimensional scatter plot for the RGB values were observed. Figure 7 depicts one such realisation. The RGB triplet of each pixel is represented as a point in this three dimensional space. Two of the chosen backgrounds represent the normal steel surface, but are chosen from different regions and varying texture while three other backgrounds consist of dust, marine growth and staining due to corrosion. It is seen that the disparate backgrounds form a well-defined dense non-overlapping cluster within the RGB space while the similar backgrounds of non-corroded steel form a single cluster. It is also important to note, that the dusty background is significantly separated from the non-corroded steel background and does not overlap with it even though they can appear visually similar on a photograph. The opinions of experts can be used as an initial interpretation and classification of the backgrounds and a comprehensive database can then be used to quantify these zones. In the absence of a more comprehensive set of field data, the authors do not attempt to quantify the exact zones associated with a certain degree of confidence. However, the clusters formed in Figure 7 can readily be used as a guideline. Table 1 has been inserted in this regard. The statistical properties of the RGB triplets are presented in Figure 8 as a Boxplot [START_REF] Mcgill | Variations of Boxplots[END_REF]. In the figure, the boxes display the median, the extremes and the quartiles in a compact way. For the purpose of comparison, the boxes for the image with HDR exposure are plotted in the standard fashion where the horizontal line within a box depicts the median value while the crosses at the ends denote the extrema. The boxes for the normally exposed image are plotted compactly, where a black dot within a circle represents the median value and the extrema are plotted at the ends. Figure 8 emphasizes the effectiveness of the methodology proposed in this paper and demonstrates how readily various surface characterisations are possible through standard statistical analysis. The values of RGB triplets from HDR image is consistently higher than that of a normally exposed image and the calibrated values for various image backgrounds are non-overlapping. Consequently, for a wide number of interpretative classes of image situations the proposed protocol is readily applicable.

Discussion and Conclusion

This paper proposes High Dynamic Range (HDR) imaging as a protocol for image processing based damage detection and Structural Health Monitoring (SHM). HDR images can capture a wide range of radiance values providing details in dark and bright, damaged and undamaged surfaces. The increased tonal range and colour value information content of HDR image over a regular photograph has been shown. HDR imaging requires no extra sensor and very little training on the part of the user but provides significantly more information than a standard imaging. The image characterisation of a number of various backgrounds, including pitting corrosion has been successfully carried out on steel specimens undergoing corrosion within a marine environment.

The proposed protocol can be applied to a range of problems and is expected to play a significant role in the specification of loads on underwater structures due to marine growth. The methodology shown in this paper can accommodate a wide array of photographic and video records from archives and provide useful information from them that can readily be used for infrastructure maintenance and management over a long period of time.

Figure 1 .

 1 Figure 1. Dynamic range of nature.

Figure 1

 1 Figure 1 illustrates this point. The column (A) shows the original dynamic range of a real-world scene and column (B) shows the same captured by a conventional digital or still camera. The actual dynamic range of the camera is shown between the dotted lines in column (B).As a result, in image-processing based SHM we are often left with photographs of significantly compromised quality for interpretation and detection purposes. The low range of brightness information of these images poses a severe limitation on what computational vision can accomplish. A very simple modification, HDR imaging, can be made to any conventional imaging system to dramatically increase this range.An HDR image is such an image whose dynamic range exceeds the dynamic range of the capturing or the display device and in turn provides more information on every pixel of the image. The HDR images with composite radiance map can be created by taking multiple photographs of the same scene with different exposures of using a standard digital camera[START_REF] Debevec | Recovering high dynamic range radiance maps from photographs[END_REF]. By using

Figure 3 .

 3 Figure 3. The increased tonal range during corrosion detection for an HDR image as compared to an image with normal exposure.

Figure 4 .

 4 Figure 4. RGB trend values for HDR image in a corrosion detection scenario as compared to an image with normal exposure.

Figure 5 .

 5 Figure 5. RGB values for an HDR image on a corroding specimen with marine growth.

Figure 6 .

 6 Figure 6. RGB values for an HDR image on a corroding specimen with dusty background.

Figure 7 .

 7 Figure 7. Clustering of RGB triplets based on background characterisation.

Figure 8 .

 8 Figure 8. Boxplots for RGB triplet values for a number of background conditions comparing HDR image based calibrations with images of normal exposure.

Table 1 .

 1 Cluster RGB values for various image backgrounds.

	Red	Green	Blue

Acknowledgement: DURATINET (Durable Transport Infrastructures in the Atlantic Area Network) project, nr 2008-1/049, approved by the "Atlantic Area


The localization of the free modes of vibration of disordered multispan beams is investigated, both theoretically and experimentally. It is shown that small deviations of the span lengths from an ideal value may have drastic effects on the dynamics of the system. Emphasis is placed on the development of a perturbation method that allows one to obtain the strongly localized modes of vibration of the disordered system without a global eigenvalue analysis of the entire system. Such a perturbation analysis is cost-effective and accurate. More importantly, it provides physical insight into the localization phenomenon, and allows one to formulate a criterion that predicts the occurrence of strongly localized modes. Also, an experiment is described which has been carried out to verify the existence of localized modes for disordered two-span beams. Theoretical and experimental results are compared in detail and excellent agreement is found, thus confirming the existence of localized modes for such weakly coupled, weakly disordered structural systems.

Nomenclature

= generalized coordinate = ith modal amplitude =peak deflection ratio =torsional spring constant =2cl/EI =unit imaginary complex number =length of beam =number of component modes of Rayleigh-Ritz procedure = Landau notation, "of the order of" =pass-band width of tuned system for jth group of modes (") ( ) '

=PBW
=angular deflection of torsional spring = .J21m1 sin hrxll), ith component mode = (i7r) 2 Ellml4, ith single-span beam frequency = w/.J Ellml 4 = (hr-)2 = natural frequency of two-span beam = n;.J EI!ml 4 =unperturbed value of 0 = first and second natural frequencies of jth group of modes =dldt =dldx

I. Introduction

T HE presence of irregularities in nominally periodic structures may localize the modes of free vibration and inhibit the propagation of vibration within the structure. This phenomenon, referred to as normal mode localization, was first predicted by Anderson in a famous paper 1 and has excited considerable interest in solid-state physics. 2 -4 Also, research studies in the field of structural dynamics 5 -13 have shown that some nearly periodic structures are highly sensitive to irregularities and may exhibit localized modes of vibration. These studies determined two categories of structural systems susceptible to localization:

1) Systems consisting of coupled, similar but slightly disordered subsystems. Typical examples include chains of coupled pendula 5 -8 and jet engine rotors, 9 for which the physical properties vary slightly from pendulum to pendulum and from blade to blade, respectively. It was shown that strong localization occurs when the coupling between subsystems is small and that localization becomes more pronounced as the coupling decreases.

2) Structures with irregularly spaced constraints. Examples include a vibrating string with irregularly spaced masses attached 6 and a beam or plate constrained at irregular intervals. 7 When strong localization occurs, small irregularities usually due to manufacturing and material tolerances result in dramatic changes in the dynamics of the system. Since neglecting these irregularities may lead to completely er-roneous results, it is particularly important to establish criteria capable of predicting the occurrence of localization.

In this paper, the strong localization of the modes of vibration of multispan beams is investigated theoretically and experimentally. Beams constrained at supposedly regular intervals are frequently encountered in structural analysis. Among numerous applications, aircraft fuselages and wings can be modeled by periodic beams. Other examples are building frames and bridges. These "periodic" structures are usually investigated by assuming ideal regularity, even though small deviations of the span lengths from an ideal value may have important effects on the free and forced response of the system.

The modes of vibration of beams simply supported at regular intervals have been studied extensively in the research literatureY- 17 Particularly, one of the first and most important contributions was made in a well-known paper by MilesY Moreover, Lin and Yang 18 investigated the effect of random deviations of the span lengths on the free modes of a multispan beam resting on simple supports, and showed that irregularities had a significant effect on the mode shapes. Nevertheless, their work was not concerned with the study of localization. Miles 14 showed that the natural frequencies of periodic multispan beams are clustered in an infinite number of groups, or bands, with n frequencies in each band, where n is the number of spans. If, in addition to a zero deflection, torsional springs exert restoring moments at the n-1 intermediate constraint locations, then the width of the frequency bands diminishes as the spring constant increases. In the limit as the spring constant c goes to infinity, the beam becomes clamped at the constraint locations, and the width of the frequency bands goes to zero.

The inverse of the torsional spring constant may be viewed as a coupling parameter between spans. As llc-0, the spans are "decoupled" because no moment can be transmitted from one span to another. For c = 0, the beam is simply supported at the constraint locations, and the spans are strongly "coupled," since no restoring moment is exerted. Hence, depending on the value of c, then-span beam can be regarded as a strongly or weakly coupled set of n-component systems. Moreover, the multispan beam is ordered if all the spans have the same length, and can be rendered disordered by considering slightly irregular constraint locations. Hence for large values of the spring constant and irregular spacing between supports, a multispan beam can be regarded as a disordered chain of weakly coupled subsystems, where each subsystem is a span. From the theory of the mode localization phenomenon, 5 • 8 • 11 the free modes of vibration of such a system are susceptible to becoming strongly localized.

In Sec. 11 of the _present paper, the modes of transverse vibration of a disordered two-span beam are investigated theoretically. It is shown that, under certain conditions, strong localization occurs. The degree of localization is dependent upon two parameters: 1) the deviation of the constraint location from the middle of the beam and 2) the value of the stiffness constant of the spring which exerts a restoring moment at the constraint location. The modes are determined by using a Rayleigh-Ritz formulation with the constraints conditions enforced by means of Lagrange multipliers. 19 The method is described in subsection A. In subsection B, classical and modified perturbation methods are presented for the analysis of nonlocalized and localized modes, respectively. These perturbation methods provide physical insight into the mechanisms of mode localization. In subsection C results are presented and discussed. These results can be readily extended to n-span beams.

Section Ill presents an experiment which has been carried out to verify the existence of localized modes for disordered two-span beams. The experiment was the primary responsibility of the second author. Both free vibration natural frequencies and spatial mode shapes were measured. The experimental apparatus is described in detail in subsection A. Subsection B presents the corresponding experimental results, along with a detailed comparison with theoretical results derived in the first part of the paper. Excellent agreement between theory and experiment is observed.

Theory

A. Free Vibration of a Disordered Two-Span Beam Consider the uniform two-span beam of length 1 shown in Fig. 1. The beam is simply supported at both ends, and is constrained to have zero deflection at x = x 1 • Moreover, a torsional spring of stiffness constant c exerts a restoring moment at x=x 1 • If x 1 = l/2, the beam is said to be tuned, or ordered; otherwise, it is mistuned, or disordered.

The equations of free bending motion are derived from Hamilton's principle, and a Rayleigh-Ritz procedure with the constraint conditions enforced by means of Lagrange multipliers is chosen. 19 The transverse deflection w(x,t) of the two-span beam is expanded in terms of the free modes of a single-span beam of length I pinned at both ends:

NM w(x,t) = E ai(t)1>i(x) i~l (1)
where ai are the generalized coordinates, and wi and 1>i (x) are the natural frequencies and normalized mode shapes of the single-span beam, respectively, defined in the nomenclature. The strain and kinetic energies of the two-span beam are (2)

1 NM T=-E af 2 i~l (3)
In addition, the beam is constrained at x = x 1 , and the two constraint equations are given by NM /1""" E ai (t)1>i (x1) = 0

(4) i==l NM !2'= E ai(t)1>f(x1)-0=0 (5)
i=l Thus the Lagrangian of the system is [START_REF] Hodges | Vibration Isolation from Irregularity in a Nearly Periodic Structure: Theory and Measurements[END_REF] where {3 1 and {3 2 are the two Lagrange multipliers corresponding to the constraints in Eqs. ( 4) and [START_REF] Hodges | Confinement of Vibration by Structural Irregularity[END_REF].

Applying Hamilton's principle, the equations of free motion are found to be [START_REF] Pierre | Localization of Vibrations by Structural Irregularity[END_REF] !2=0 [START_REF] Valero | Vibration Characteristics of Mistuned Shrouded Blade Assemblies[END_REF] Assuming simple harmonic motion of natural frequency, n, one has i= 1, ... ,NM; k=1,2

For !l~wi, Eq. ( 7) may be written as [START_REF] Bendiksen | Mode Localization Phenomena in Large Space Structures[END_REF] Substituting the above expression of a; into Eqs. ( 8) and ( 9) yields, for O~w;

Nonzero solutions are obtained for ~1 and ~2 if and only if the determinant of the system [Eqs. [START_REF] Klosterman | Modal Surveys of Weakly Coupled Systems[END_REF] and [START_REF] Craig | Modal Vector Estimation for Closely-Spaced-Frequency Modes[END_REF]] is equal to zero, yielding

where w; and fi are dimensionless frequencies.._£ is the dimensionless spring constant, and .X 1 =x 1 11= Y2-ill is the dimensionless location of the intermediate support, where ill is the dimensionless deviation from the middle of the beam.

Recall that Eq. ( 14) presupposes fi ~ w;. It is an eigenvalue equation whose solutions are the free vibration natural frequencies fi of the two-span beam. For each value of fi solution of Eq. ( 14), the corresponding ratio ~1 /~2 is obtained from either Eq. ( 12) or Eq. ( 13), and the generalized coordinates amplitudes a; are given by Eq. ( 11), from which the expression of the spatial mode shape w is readily obtained from Eq. ( 1).

Tuned Beam

In this case ill= 0. The natural frequencies of a tuned beam simply supported at its middle are [START_REF] Lin | Free Vibrations of Continuous Beams on Elastic Supports[END_REF] (16)

These natural frequencies have a pass-band character, and are placed in groups of two along the frequency axis. 14 As c increases, the first frequency of the group increases, while the second remains unchanged. For a beam clamped in the middle, the two natural frequencies of each group are equal, leading to twofold multiple eigenvalues. Hence the width of the frequency bands decreases and goes to zero as c goes to infinity. As will be shown later, this bandwidth is one of two key parameters in determining the occurrence of localized modes.

Mistuned Beam

If, for a given value of c, a mistuning ill is introduced, the two frequencies of a group move apart: the width of the frequency bands increases with M. This behavior is shown in Fig. 2, which represents the first and second natural frequencies (first group of modes) in terms of ill for various values of c. It should be noted that, for relatively large values of M such as 0.07, the band character of the natural frequencies is lost.

Convergence

The convergence of the Rayleigh-Ritz procedure with the number of component modes NM has been checked by considering a mistuned beam clamped at .X 1 , hence defined by M~ 0 and c-oo. In this case the natural modes are those of the two hinged-clamped spans of lengths Y2 -M and Y2 +ill, and thus the exact mode shapes should have exactly a zero deflection over one of the spans. It was found that a large number of component modes must be considered in order to achieve good convergence. Typically, the mode shapes were almost perfectly flat over one of the spans if NM~ 1000. In the subsequent calculations, 1000 component modes were used. Convergence was also checked for higher modes: until the 25th mode at least, nearly zero deflection in one of the spans was obtained if 1000 or more component modes were used.

This rather slow convergence can be explained by noticing that the series E;':,"{(l!i 4 ) and E):,"{ (liP) are involved in the eigenvalue Eq. ( 14). Although the former series quickly converges, the latter is slowly convergent, thus the Rayleigh-Ritz expansion adopted here slowly converges to the exact solution. However, since this is a linear calculation, the computer cost remains reasonably low. Moreover, if the number of component modes is large enough, very accurate results are obtained, and higher modes can be calculated as easily and accurately as lower ones.

B. Perturbation Analyses

It is much easier to calculate the modes of a tuned twospan beam than of a mistuned one, because the eigenvalue equation is significantly simpler in the former case. This is characteristic of nearly periodic structures with small irregularities: when the structure is disordered, its periodicity properties are lost, and investigating its modes of vibration requires a computational effort much greater than for the associated periodic system. Hence the idea, for small irregularities, of performing a perturbation analysis.

Classical Perturbation Analysis

The unperturbed system consists of the tuned beam. It is perturbed by moving the constraint by a distance ill. A perturbation analysis can be readily defined from Eq. (14l_ by expanding the terms sin{i7r.X 1 ) and cos(i7r.X 1 ) in terms of M to the first or second order, hence obtaining the corresponding natural frequency perturbations ofi and o 2 fi. This perturba-100.

z Jtw tion analysis is straightforward and will not be presented in detail. The method has been previously used by Lin and Yang 18 for a beam on simple supports. Note that this approach presupposes that the term lie in Eq. ( 14) is not small, but has a finite or large value. For 11 c finite, the modes of vibration of the mistuned system are perturbations of (hence are very similar to) the modes of the tuned system, and are certainly not localized. Thus this case is not of great interest to the present study.

0,~---------~~~~--------~-•~x.

Modified Perturbation Analysis

Herein, small values of 11 c are considered. If, when the system is mistuned, only fll is considered as a perturbation, then one can expect qualitatively erroneous results: all the small parameters (not only M, but also 11 c) must be treated as perturbations. The unperturbed system would then be characterized by fl/=0 and llc=O, defining a beam clamped at its middle. The perturbations would consist of fll and 11 c, leading to a mistuned, "almost" clamped beam. The natural frequencies of the unperturbed system are then repeated, with one twofold multiple eigenvalue in each group. The corresponding mode shapes are defined by any linear combination of a left-span hinged-clamped mode and of a right-span clamped-hinged mode, since the eigenfunctions associated with each double natural frequency span a space of dimension two. In order to perform a perturbation analysis, one must first determine the unique set of two unperturbed mode shapes from which the modes of the perturbed system are continuously obtained. It can be shown 20 that this is equivalent to solving the eigenvalue problem for the modes of interest, hence rendering this perturbation procedure ineffective. The conclusion is that one must avoid multiple eigenvalues for the unperturbed system. This is achieved by introducing some mistuning in the unperturbed system. The unperturbed state is then defined by 11 c = 0 and i 1 = Yzfll. It consists of a mistuned twospan beam clamped at the constraint location. The only perturbation parameter is 11 c. This perturbation method is referred to as modified perturbation method (MPM), and is similar to the one developed in Ref. 8 for a disordered chain of weakly coupled pendula. Since the unperturbed beam is mistuned, its eigenvalues are simple. Also, since it is clamped at i =.X\, its natural modes are th~ ones of hinged-clamped beams of lengths Yzfll and Yz + fll. Note that these unperturbed modes are decoupled, that is, they have a zero deflection over one of the two spans, depending on the mode number. When the system is perturbed by 11 c, the modes cease to be decoupled in the left or right spans, and have nonzero deflection in both spans. However, since 11 c is small, they are perturbations of the decoupled modes, hence are characterized by a deflection which is much larger in one span than in the other one: the modes are strongly localized.

It is remarkable that one is able to predict whether the modes are localized or not, just by considering perturbations of the eigenvalue Eq. ( 14).

Let 0 0 be a natural frequency of the unperturbed system.

The system is perturbed by replacing the clamped condition by a spring of high stiffness c, and the natural frequencies n of the perturbed system are such that [START_REF] Mead | Vibration Response and Wave Propagation in Periodic Structures[END_REF] where llO is a first order perturbation in 11 c. Substituting this first order expansion into Eq. ( 14) and expanding to the first order yields (18a)

where

NM +E i~I (i7r-)2 cos 2 (i1ri 1 ) w7-fi~ (18b)
The corresponding perturbed mode shape is readily calculated by substituting the perturbed value of 0 into Eqs. [START_REF] Anderson | Absence of Diffusion in Certain Random Lattices[END_REF] and [START_REF] Bendiksen | Mode Localization Phenomena in Large Space Structures[END_REF][START_REF] Klosterman | Modal Surveys of Weakly Coupled Systems[END_REF][START_REF] Craig | Modal Vector Estimation for Closely-Spaced-Frequency Modes[END_REF]. Note that a second-order perturbation analysis can also be easily developed. Two important characteristics of perturbation methods are retained by the present analysis. First, the method is cost effective. Second, strongly localized modes are predicted for small values of 11 c if the beam is mistuned: perturbation methods provide physical insight into mode localization.

C. Results and Discussion

Results

For given values of fll and c, the eigenvalue equation [Eq. [START_REF] Miles | Vibration of Beams on Many Supports[END_REF]] is solved by a standard bisection technique. The bisection process converges rapidly. Typically, 20 to 35 iterations are necessary to obtain natural frequencies converged up to the lOth decimal place. This kind of accuracy is required because very small variations in the natural frequencies may result in significant variations in the mode shapes, since a large number of component modes are considered. The accuracy of the Rayleigh-Ritz procedure and of the bisection process was checked against known results, and in all cases excellent agreement was observed. Even though the number of component modes considered is very large, the CPU time necessary was not excessive. Unless otherwise stated, the following results were obtained by solving directly Eq. ( 14), not by perturbation analysis.

Figure 3 shows the lower two modes of a tuned beam (fll=O) such that c= 1000. One observes that the modes are collective, as opposed to localized: the magnitude of the deflection is the same in each span. Figure 4 displays the lower two modes of a mistuned beam such that fl/=0.01, for the same c= 1000. One clearly sees that the peak deflection is much larger in one span than in the other one: slight mistuning is sufficient to localize strongly the natural modes. The localized modes are perturbations of the "decoupled" modes corresponding to llc=O. Since the system is mistuned, these decoupled modes correspond to simple eigenvalues. On the other hand, the modes of the tuned system such that 11 c = 0 correspond to twofold multiple eigenvalues, and perturbed modes for small 11 c do not vary continuously from individual decoupled modes, giving rise to collective modes. The results shown in Fig. 4 were obtained by both the exact method and the modified perturbation method. The agreement is excellent, confirming the fact that the MPM is suitable for the analysis of strongly localized modes. If the spring constant c increases, the mode shapes become even more strongly localized. This is observed in Fig. 5, which displays the lower two modes for c = 5000, and for the same mistuning parameter fll=O.Ol. In the limit c-oo, the modes of the mistuned beam tend to become decoupled. On the other hand, for larger values of 11 c, the modes are only partially localized, and to the limit c-0, the modes of the {gmply supported) mistuned beam are not localized. For fll=O.Ol and c=O, it was found that there is only a slight difference between the peak deflections in each span. In this case, the mode shapes are no longer perturbations of decoupled modes, but are perturbations of the collective modes of the tuned beam, which are shown in Fig. 3. Hence the classical perturbation method (defined for large values of 1/ c) would be suitable for this analysis.

In order to investigate systematically the effect of the mistuning parameter tJ./ and of the spring constant c, it is suitable to adopt a compact representation of the modes. The degree of localization of a mode can be characterized by the ratio A of the peak deflection in one span to the peak deflection in the other span, such that the numerator of this ratio corresponds to the span with the smaller peak deflection [START_REF] Dowell | Free Vibrations of an Arbitrary Structure in Terms of Component Modes[END_REF] where As and At are the peak deflections in each span, such that As ~At. Note that the ratio A takes values ranging be- o:

.03 .02 0.05 Jr-----~-. . . v ooL----t-~-.. ~----+-1 --t-1 ---<,__t-1 ---+-1 _ ____.~ 0 200. 400. 600. 800. 1000.

I

Fig. 6 Values of lA I in the (c,..il) plane, for the first group of modes.

tween -1 and + 1. The smaller the absolute value of A, the more localized the corresponding mode. For decoupled modes, A= 0. For a tuned beam, no matter how large c is, A = ± 1, depending on the mode number. Figure 6 displays values of lA I in the (c,tJ.l) plane, for the first group of modes. To fix ideas, localization is said to occur if the absolute value of the peak ratio A is less than lOOJo. Note that for a given c, A decreases as tJ./ increases, hence the mode localization becomes more pronounced as the amount of mistuning is increased. In the limit c-oo, the modes are localized for an arbitrarily small, but nonzero, mistuning. Also, for a given mistuning tJ.l, localization becomes more pronounced as c increases. The larger M, the smaller the threshold value of c necessary to give rise to localized modes. However, strong localization does not occur for c< 110, even for relatively large values of mistuning M such as 0.07. In particular, the lower modes of a beam simply supported at the constraint location do not become strongly localized. Even if the value 0.07 seems to be small, the reader should bear in mind that this study is conducted within the context of small perturbations, and that tJ./ = 0.07 corresponds to a 14% deviation of the length of the individual spans, a fairly large value. An approximate boundary of localization, corresponding to lA I = lOOJo, is represented on Fig. 6 by a dotted line.

From numerical results, one can show that, for various small values of 11 c and t.l such that the product ct:J./ is the same, the peak ratio A remains approximately constant. This can be seen in Fig. 6, as the localization boundary is similar to a hyperbola of equation c = const/ t:J.l. The product et./ is in fact a disorder to coupling ratio t.//(1/c), and the degree of localization seems to depend only on the value of this ratio. This result is similar to the one obtained in Refs. 5 and 8 for a chain of coupled pendula. Nevertheless, in Ref. 8, this was shown analytically, whereas in the present study, one is required to investigate numerically the dependence of A upon t:J.l and c.

Discussion

There is a strong analogy between the two-span beam and the system of two coupled pendula studied in Ref. [START_REF] Pierre | Localization of Vibrations by Structural Irregularity[END_REF]. The two pendulum system consists of two coupled single DOF oscillators, each of them being characterized by an individual natural frequency. The amount of coupling is governed by the value of the spring constant k, and mistuning is achieved by changing slightly the individual natural frequencies of the pendula. Similarly, for the two-span beam, the coupling between spans is determined by the inverse of the torsional spring constant. If 11 c = 0, the spans are "decoupled," the same way the two pendula are decoupled for k=O. Each of the individual spans possesses an infinity of natural frequencies, which are for hinged-clamped boundary conditions. Hence if the beam is tuned (respectively mistuned), the two spans have identical (respectively different) individual natural frequencies. It should also be noted that the twospan beam is a system of two coupled, infinite number of DOF oscillators, whereas the pendulum system is constituted of single DOF oscillators. However, recall that the natural frequencies of a two-span beam are distributed by groups of two, and each of these groups can be regarded as corresponding to a two-pendulum system. It has been shown in Refs. 5 and 8 that the modes of the pendulum system are strongly localized for small values of mistuning and coupling. Similarly, for the two-span beam, strong localization occurs for small values of t:J.l and 11 c. For the pendulum system, localized modes are perturbations of decoupled oscillations; for the two-span beam, they are perturbations of "decoupled" hinged-clamped modes. It has also been shown in Ref. 8 that strong localization does not occur for strong coupling between pendula. Similarly, the modes of a two-span beam are not localized for finite or large values of 11c. In particular, strong localization does not occur for c = 0, even for relatively large values of t:J./ such as 0.07. Moreover, it is clearly seen in Fig. 6 that the effect of mistuning on the peak ratio is not drastic for c = 0, but rather slowly increasing with t:J./. On the other hand, for larger values of c, a rapid change of A in terms of t:J./ is observed: strong localization occurs. To conclude, the theory of the mode localization phenomenon is for small departure from ideal regularity. However, for larger values of mistuning, even though the modes do not become strongly localized, significant chan_gt:s can also be observed. For instance, in the case c = 0 and t.l = 0. 07, one observes from Fig. 6 that the peak ratio A of the first group of modes is 0.4, significantly different from the tuned case. Although not relevant to the study of strongly localized modes, these changes may be of interest and of potential importance to the designer. It should also be mentioned that the analogy between the two-pendulum system and the two-span beam can be readily generalized to an n-pendulum system and an n-span beam, for any n, suggesting that localized vibrations also occur for multispan beams.

In order to understand thoroughly the physical mechanisms of localization for multispan beams, the criterion formulated in Ref. 8 is now considered. This criterion states that a disordered chain of coupled oscillators is susceptible to having strongly localized modes if the natural frequencies of the corresponding tuned, or periodic, system are distributed in groups, and if the widths of these pass-bands are small relative to the values of the frequencies belonging to the pass-bands. Localization may occur for such systems if a characteristic spread (due to mistuning) in the individual frequencies of the component subsystems is small, and of the order of, or larger than the pass-band width of the ordered system

PBW::;fJ(SNF) (20)
of the corresponding tuned, or periodic, system are distributed in groups, and if the widths of these pass-bands are quencies. Note that for a tuned, or periodic system, SNF= 0. As mentioned previously, the natural frequencies of a tuned multispan beam have a pass-band character. 14 Denoting the pass-band width of the jth pair of modes, which contains the (2}-l)th and 2jth modes, by PBWj, one can write (21) where the natural frequency of the 2jth mode is given by Eq. (16). Since 0 2 j -l (c) increases with c, PBWj(c) decreases as c increases. For 1/c small, PBWj(c) is approximately a linear function of lie, and in the limit 1/c-0, PBWj goes to zero. Hence, small values of the "coupling" 11 c mean small pass-band width of the ordered system.

The other variable that needs to be defined is the SNF. As previously stated, the beam is decoupled if 11 c = 0, its natural frequencies being the ones of the two individual hinged-clamped spans. The spread resulting from mistuning can be written as

I rm - -<- 1 -±-2 ~~= 1 ~>-2 =-...} ---;;;;!4 n2j4 1 ~~~ (22)
Hence, for small mistuning, SNFj is proportional to the amount of mistuning.

The following discussion investigates the ability of the criterion [Eq. ( 20)] to predict localized modes. This paragraph is concerned with the first group of modes, corresponding to j = 1. Figure 7 displays the absolute value of the peak ratio lA I in the (PBW 1 ,SNF 1 ) plane, for the first group of modes. Note that PBW 1 and SNF 1 have been nondimensionalized by 0 2 , as defined in the nomenclature. It is observed that localization occurs when SNF 1 and PBW 1 are both small. Moreover, with the definition of localization lA I ::; lOOJo, the modes are localized in the region approximately defined by PB W 1 ::; 0.42 SNF 1 , the localization boundary being given by PB W 1 = 0.42 SNF 1 • Note that this boundary is dependent upon the definition chosen for localization: stronger or weaker requirements for localization to occur would result in a quantitatively different, but qualitatively similar boundary. It should also be noted that, from numerical results, the degree of localization lA I seems to be only dependent upon the ratio PB W 1 I SNF 1 • Since the localization region shown in Fig. 7 is consistent with the criterion [Eq. ( 20)], the latter has the ability to predict the occurrence of localization for the first group of modes.

Finally, it is of interest to investigate the localization of higher groups of modes. From Eq. ( 22), it is clear that SNFj remains constant when j increases. Also, for a given c, PBWj decreases as j increases, and goes to zero in the limit j-oo [for example this can be seen from Eqs. ( 15) and(! §) for c = 0]. Thus for any given c (even small), and a given ~/, there exists a group number j* such that PBWj is smaller than SNFj, for any}> j*. According to Eq. (20), this would mean that, for any c and ~!, no matter how small, there always exists a threshold value j* such that higher groups of modes are localized. However, preliminary results do not seem to confirm this hypothesis.

Figure 8 shows the variation of the peak ratio lA I in terms of the mode number, for various values of c and ~/.

For c= 1000 and M= 0.01, localization occurs in the first group of modes. Higher modes are still localized, but no more strongly than the first two modes. As a matter of fact, the peak ratio remains almost constant when the mode number increases. For c= 110 and M=0.019, the first group of modes is not localized, and the peak ratio decreases only slightly in the higher modes, from 0.33 for the first group to a plateau value of 0.26 for the sixth group. Finally, for c=O and ~~ = 0.03, the peak ratio decreases significantly from 0.64 for the first group to 0.42 for the fifth group. However, the modes do not become localized. Moreover, after the lOth mode, lA I increases to reach 0.93 in the eighth group of modes, and goes back to 0.49 for the 20th mode. In this case, PBWj decreases monotonically and one can show that according to Eq. ( 20) localization ought to occur in the eighth or ninth group of modes. However, it does not. Two hypotheses can be formulated from the study of these few representative cases: 1) If the modes of the first group are not localized, it seems that localization will not occur for the modes of higher groups either.

2) If the first two modes are localized, then higher modes are also localized, but no more strongly than in the first group.

This suggests that higher modes do not behave significantly differently than lower modes with respect to localization. Hypothesis 2 is a reassuring result, since it states that localization does not disappear in higher modes. Hypothesis 1 is, of course, in a sense disappointing. These preliminary results suggest that the criterion [ Eq. (20)] cannot be used independently of the physical system to which it applies: the most important condition for strong localization to occur is to have a weakly coupled system, that is to have 11 c small.

If this requirement is met, then the criterion [ Eq. (20)] can be applied effectively to determine the minimum value of mistuning M necessary to obtain strongly localized modes. Finally, one ought to mention that, even though the higher modes do not become strongly localized for c = 0, mistuning may have a significant effect, since for ~~ = 0.03 the peak ratio of the seventh mode is 0.40.

Ill. Experiment

Few experimental studies of localized vibrations have been conducted to date. Hodges and W oodhouse 6 carried out an experiment to demonstrate localization, and found satisfactory agreement with the theoretical predictions. The system used in the experiment was a stretched string with irregularly spaced masses attached to it. Also, the research study by Craig et al. 13 evidenced strong discrepancies between experimental and analytical results for a system of two weakly coupled beams. It was shown in Ref. 13 that these discrepancies were due to small physical dissimilarities between the two beams. The present authors believe that the system studied in Ref. 13 did indeed exhibit strongly localized modes.

A. Experimental Apparatus

The modes of free vibration of a two-span beam were investigated experimentally. The vibration tests were performed on a spring steel beam resting on three supports. The beam was pinned at both ends. In addition, a third support with variable torsional stiffness was located near the middle of the beam, but could be moved to various locations. The experimental apparatus is shown in Fig. 9. The geometric dimensions of the specimen beam were 53 cm (length) and 0.0635 x 1.015 cm 2 (cross section). The variable torsional stiffness of the intermediate support was created by a pinned-clamped beam, the distance between the pinned point and the clamped one being varied to adjust the torsional stiffness. The torsional beam was parallel to the specimen beam (see Fig. 9).

The possibility of dynamic interaction between the specimen and torsional beams had to be considered. The frequency range of the first and second modes of the specimen beam was 19-40 Hz. The dynamic torsional stiffness of the torsional beam was measured near the pinned point when removing the specimen beam. It was found that the torsional stiffness remained essentially constant in the frequency range of interest . In general, when the fundamental natural frequency of the torsional beam is two to three times higher than the one of the specimen beam, the dynamic torsional stiffness does not vary significantly, and thus can be considered to be constant. In our experimental apparatus, the torsional beam frequency was more than ten times higher than the specimen beam frequency.

The major equipment components for the vibration test were as follows:

1) A sine generator provided a sweeping sinusoidal signal.

2) A minishaker (B&K 4810) and a power amplifier (B&K 2706) were used to excite the beam. In order to reduce the effect of the additional mass of the joint components between the shaker and the beam on the vibration characteristics of the specimen beam, the driving point was located near the pinned end of the torsional beam (see Fig. 9). Thus a pure excitation torque was applied to the specimen beam near its intermediate support, without inducing any appreciable added mass effect. Because of the small mass of the specimen beam, this effect could have been potentially very important.

3) A force transducer (B&K 8200) and two rotaryvariable-differential transformers (R30D) were used to measure the exciting force and transverse beam displacement, respectively. The displacement transducers had only a small added mass.

4) The charge amplifier was a portable conditioning amplifier (B&K 2635) which provided high-voltage output sensitivity of the force. Digital voltage meters were used to record all signals and to analyze natural frequencies and mode shapes.

In addition to the effect of added mass, there were two other important considerations in the design of the experiment. The first one was concerned with minimizing the effect of the additional constraint due to jointing the R30D transformers with the specimen beam. In order to avoid additional stiffness constraint when large-amplitude vibration occurs, the contact between the needle and the beam had to be sufficiently flexible. This requirement was met by using a flexible needle with a pinned end.

The second consideration concerned the design of the pinned end supports of the specimen beam. From a transient decay test, the critical damping ratio of the beam was found to be approximately 0.001. Thus, large amplitudes occurred near the resonance frequencies. If the horizontal displacement of the end supports were constrained, the measured frequency was found to be dependent upon the level of the excitation torque, which is characteristic of a nonlinear system. Thus, in the experimental apparatus, two degrees of freedom, namely rotation and horizontal displacement, were allowed at the pinned ends, in order to ensure the linearity of the system. It was then found that the natural frequencies were independent of the excitation torque level, and that even when the response amplitude was very large, the system behaved in a linear fashion.

B. Experimental Results: Comparison with Theory

The displacement mobility concept (response displacement/excitation torque) was used to determine the natural frequencies. Figure 10 shows the dependence of the lower two natural frequencies upon torsional stiffness for a tuned beam (M= 0), for both experimental and theoretical results. The agreement between theory and experiment is observed to be excellent. Note that the torsional stiffness was determined from a static stiffness measurement.

Figure 11 displays the comparison between theoretical and experimental natural frequencies versus mistuning M, for a coupling c= 281.8. Again, the agreement is found to be ex- cellent. Lack of space precludes the authors from presenting results obtained for other values of the coupling c. Nevertheless, in all cases studied, the maximum discrepancy between theoretical and experimental results was always less than 2.50Jo.

Very good agreement was also found between theoretical and experimental results in terms of mode shapes. This can be observed on Figs. 12a and12b, which display the peak ratio A for the lower two modes in terms of mistuning A/, for values of the torsional spring constant c = 90.4 and c= 281.8, respectively. Peak ratios were also compared for other values of c, but these results are not presented here.

The maximum difference between theoretical and experimental data was always less than 15%. This error was mainly due to inaccuracies in the measurement of the small response amplitudes which were encountered for strongly localized modes. For at very small amplitudes the signal to noise ratio of the transducers R30D becomes smaller.

Finally, Fig. 13 shows the motion in the first mode for a torsional spring constant c = 281.8. Figure 13a is for the tuned system, whereas Fig. 13b is for a slightly mistuned beam such that A/= 2%. Both are obtained from displacement measurement. It is observed that the first mode of the mistuned beam is strongly localized in the second span, whereas the one of the tuned beam is collective, that is the peak deflection is the same in both spans.

IV. Concluding Remarks

The modes of vibration of disordered two-span beams subject to a restoring torsional spring moment at the intermediate support have been investigated theoretically and experimentally. The following conclusions can be drawn:

1) For small mistuning and large torsional spring constant, the modes of vibration become strongly localized in one of the two spans.

2) A modified perturbation method has been developed. It predicts strongly localized modes accurately and provides physical insight into mode localization.

3) For the first group of modes, strong localization occurs if the relative pass-band width of the tuned beam is of the order of, or smaller than the relative spread in the frequencies of the individual spans, and if these two quantities are both small. 4) From preliminary results, it is suspected that if localization does not occur in the lower two modes, then it does not occur in the higher ones either. On the other hand, if the first two modes are localized, then the higher ones are also localized.

5) An experiment has been carried out to verify the existence of strongly localized modes for disordered two-span beams. Excellent quantitative agreement has been found with theoretical results.

6) An immediate generalization of the present study is to investigate the localization of vibrations for n-span beams, where n > 2. Since disorder is usually caused by uncertainties, a statistical approach will be required. Future work is also in order concerning localized vibrations of multispan beams resting on elastic supports (leading to multicoupling between spans) and two-dimensional structures.

/fl 2 j

 2 =spread in natural frequencies for jth group of modes =SNF/fl 2 j

Fig. 1

 1 Fig. 1 Geometry of disordered two-span beam.

Fig. 3

 3 Fig.3Lower two mode shapes for a tuned two-span beam (M= 0) for c= 1000.

Fig. 4

 4 Fig.4 Lower two mode shapes for a mistuned two-span beam, for ~ = 0.01 and c = 1000, by exact method and modified perturbation method.

5 . 4 :

 54 Lower two mode shapes for a mistuned two-span beam, for 4/=0.01 and c=SOOO. -~5 14 I .~6

1 Fig. 7

 17 Fig. 7 Values of lA I in the (PBW 1 , SNF 1 ) plane, for the first group of modes.

Fig. 9 Fig. 10

 910 Fig. 9 Experimental apparatus.

Fig. 11 Fig. 12

 1112 Fig. 11 Comparison of experimental and theoretical natural frequencies of the first group of modes, for c = 281.8.

Fig. 13a

 13a Fig. 13a First mode shape of tuned two-span beam, for .1/ = 0 and c = 281.8, from measurements.

Vibrations of Disordered Multispan Beams: Theory and Experiment

	Localized Christophe Pierre
	University of Michigan, Ann Arbor, Michigan
	and
	De Man Tang, and Earl H. Dowell
	Duke University, Durham, North Carolina
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Acknowledgments

This work was supported by the AFOSR under Grant 85-0137 with Duke University. Dr. Anthony Amos is the technical monitor.