Lindita Imeraj
email: lindita.imeraj@ugent.be

Inge Antrop

Herbert Roeyers

Ellen Deschepper

Sarah Bal

Dirk Deboutte

Diurnal variations in arousal: a

Keywords: ADHD, arousal, circadian, heart rate, activity

come

INTRODUCTION

Attention-Deficit Hyperactivity Disorder (ADHD) is one of the most common child psychiatric diagnoses typified by inattention problems, hyperactivity and impulsivity (American Psychiatry Association, 2000) [START_REF]Diagnostic and statistical manual of mental disorders[END_REF]. Several theoretical models have explained the behavioral symptoms in this developmental disorder in terms of arousal regulation deficits [START_REF] Sergeant | Convergence of approaches in localizing the hyperactivity deficit[END_REF][START_REF] Sergeant | What happens after a hyperactive child commits an error?[END_REF][START_REF] Zentall | Optimal stimulation -a model of disordered activity and performance in normal and deviant children[END_REF]. As arousal is closely related to sleep-wake regulation [START_REF] Schwartz | Neurophysiology of sleep and wakefulness: basic science and clinical implications[END_REF], it is not surprising that arousal is characterized by a circadian rhythm [START_REF] Silver | Circadian and homeostatic factors in arousal[END_REF]. To date, several research domains have pointed to differential circadian arousal patterns in ADHD. There is evidence that subjects with ADHD show a higher eveningness preference [START_REF] Caci | Inattentive symptoms of ADHD are related to evening orientation[END_REF][START_REF] Rybak | Seasonality and circadian preference in adult attention-deficit/hyperactivity disorder: clinical and neuropsychological correlates[END_REF][START_REF] Susman | Morningness/eveningness, morning-to-afternoon cortisol ratio, and antisocial behavior problems during puberty[END_REF], more sleep-wake disturbances [START_REF] Boonstra | Hyperactive night and day? Actigraphy studies in adult ADHD: a baseline comparison and the effect of methylphenidate[END_REF][START_REF] Cortese | Sleep in children with Attention-Deficit/Hyperactivity Disorder: Meta-analysis of subjective and objective studies[END_REF][START_REF] Hvolby | Actigraphic and parental reports of sleep difficulties in children with attention-deficit/hyperactivity disorder[END_REF][START_REF] O'brien | Sleep and neurobehavioral characteristics of 5-to 7-year-old children with parentally reported symptoms of attention-deficit/hyperactivity disorder[END_REF][START_REF] Owens | Subjective and objective measures of sleep in children with attention-deficit/hyperactivity disorder[END_REF], and more hyperactivity problems in the afternoon [3, [START_REF] Porrino | A naturalistic assessment of the motor-activity of hyperactive boys. 1. Comparison with normal controls[END_REF]60,67]. As findings on this research topic are inconsistent [START_REF] Corkum | Actigraphy and parental ratings of sleep in children with attention-deficit-hyperactivity disorder (ADHD)[END_REF][START_REF] Dagan | Sleep quality in children with attention deficit hyperactivity disorder: an actigraphic study[END_REF][START_REF] Gruber | Instability of sleep patterns in children with attentiondeficit/hyperactivity disorder[END_REF][START_REF] Mick | Sleep disturbances associated with attention deficit hyperactivity disorder: The impact of psychiatric comorbidity and pharmacotherapy[END_REF], further clarification of time of day effects in arousal is critical to fully understand underlying mechanisms involved in arousal and behavioral symptoms of ADHD.

One of the major systems involved in arousal regulation is the autonomic nervous system [START_REF] Pfaff | A theoretical framework for CNS arousal[END_REF][START_REF] Schwartz | Neurophysiology of sleep and wakefulness: basic science and clinical implications[END_REF][START_REF] Silver | Circadian and homeostatic factors in arousal[END_REF].

Heart rate has been considered as a reliable, non-invasive measure for the autonomic activity of an individual [START_REF] Mezzacappa | Anxiety, antisocial behavior, and heart rate regulation in adolescent males[END_REF][START_REF] Scheer | Cardiovascular control by the suprachiasmatic nucleus: Neural and neuroendocrine mechanisms in human and rat[END_REF], which is especially related to emotional and motivational aspects of arousal [START_REF] Arnett | Gray's three-arousal model: an empirical investigation[END_REF][START_REF] Groeppel-Klein | Arousal and consumer in-store behavior[END_REF][START_REF] Oldehinkel | Low heart rate: A marker of stress resilience. The TRAILS study[END_REF]. Heart rate reflects the balance between the activity of the two branches of the autonomic nervous system, the sympathetic and parasympathetic division [START_REF] Oldehinkel | Low heart rate: A marker of stress resilience. The TRAILS study[END_REF], which are associated with stress and rest states respectively [START_REF] Dietrich | Externalizing and internalizing problems in relation to autonomic function: A population-based study in preadolescents[END_REF][START_REF] Jose | Effect of combined sympathetic and parasympathetic blockade on heart rate and cardiac function in man[END_REF]. This parasympathetic branch, more specifically the ventral vagal complex, is also involved in emotion and behavior regulation processes such as attention, motor activity, emotion, and social engagement [START_REF] Porges | Orienting in a defensive world: Mammalian modifications of our evolutionary heritage. A polyvagal theory[END_REF][START_REF] Porges | The polyvagal perspective[END_REF].

To date however it is unknown whether circadian regulation of the autonomic nervous system is different in children with ADHD as no evaluation of 24-hour heart rate patterns is available. One study with a shorter-term evaluation (three periods of 5 min) suggested that children with ADHD have elevated heart rate levels compared to controls [START_REF] Tonhajzerova | Changes in the cardiac autonomic regulation in children with attention deficit hyperactivity disorder (ADHD)[END_REF]. This finding was associated with changes in cardiac autonomic regulation, i.e., decreased cardiac vagal modulation in supine position and altered ability of dynamic activation of the autonomic nervous system in response to orthostasis.

The main aim of this paper is to investigate whether 24-hour heart rate patterns under natural environmental conditions are different in non-medicated children with ADHD as compared to normal control children. We have a particular interest in resting heart rate, i.e., nighttime levels, as this measure can be used as an indicator of autonomic balance and vagal tone [START_REF] Thayer | The role of vagal function in the risk for cardiovascular disease and mortality[END_REF]. Based on a short-term evaluation study [START_REF] Tonhajzerova | Changes in the cardiac autonomic regulation in children with attention deficit hyperactivity disorder (ADHD)[END_REF], we expect elevated heart rate levels, i.e., a decreased vagal tone, in children with ADHD as compared to normal controls. In line with previously reported circadian alterations in arousal, we further hypothesize that group differences in heart rate are particularly expressed at specific times of day.

Additionally, we take into account several possible confounders of heart rate levels. First, demographic variables such as age and sex, but also pubertal stage (Tanner) and body mass index (BMI) could influence heart rate levels [5,[START_REF] Wallis | Age related reference ranges for respiration rate and heart rate from 4 to 16 years[END_REF]. In this context, also average use of caffeine and presence of nightmares were questioned.

Second, both externalizing problems such as oppositional defiant disorder (ODD) and conduct disorder (CD) and internalizing problems such as anxiety and depression have been related to different autonomic functioning, i.e., lower and higher resting heart rate levels respectively [START_REF] Dietrich | Externalizing and internalizing problems in relation to autonomic function: A population-based study in preadolescents[END_REF][START_REF] Lorber | Psychophysiology of aggression, psychopathy, and conduct problems: A meta-analysis[END_REF][START_REF] Mezzacappa | Anxiety, antisocial behavior, and heart rate regulation in adolescent males[END_REF][START_REF] Ortiz | Heart rate level and antisocial behavior in children and adolescents: A meta-analysis[END_REF]. Third, activity is hypothesized to influence heart rate levels. We simultaneously assess 24-hour activity patterns in order to explore possible confounding effects on heart rate patterns and to guide interpretations on the behavioral level. We suggest a close relation between activity and heart rate levels, and expect higher activity levels in children with ADHD as compared to normal controls, especially during afternoon hours.

METHOD

Subjects

Thirty children diagnosed with ADHD (24 boys and 6 girls), aged 6-11, were recruited from a child psychiatric outpatient unit. As controls, we selected 30 age-and sex-matched healthy, normally developing children within the same class as the child with ADHD. Control children had no formal psychiatric or medical diagnosis, and were all medication-free. Children with ADHD had a formal diagnosis of ADHD combined typeboth inattention and hyperactivity/impulsivity problems-based on a child and adolescent psychiatric evaluation during which also a structured interview was used to assess symptoms of ADHD in accordance with the Diagnosis and Statistical Manual of Mental Disorders criteria (Diagnostic Interview Schedule for Children for DSM-IV, parent version (PDISC-IV)) [START_REF] Shaffer | NIMH Diagnostic Interview for Children version: Description, differences from previous versions and reliability of some common diagnoses[END_REF]. Exclusion criteria included an IQ less than 80 as measured by the Wechsler Intelligence Scale for Children, third edition (WISC-III-R) [START_REF] Wechsler | Wechsler intelligence scale for children[END_REF], pervasive developmental disorders, neurological disorders such as epilepsy, chronic medical conditions such as hypertension, diabetes mellitus, and asthma, and any medical treatment (e.g., atomoxetine or psychotropic medications) except for methylphenidate. 26 children with ADHD took methylphenidate (17 short-and 9 long-acting formulation), but all children were free from treatment at least 72 hours prior to participation in the study. None of the participating children used alcohol or nicotine. Current behavioral and emotional problems were screened in both groups using the Child Behavior Checklist (CBCL) [START_REF] Achenbach | Child Behavior Checklist for ages 4-18 (CBCL/4-18)[END_REF][START_REF] Verhulst | Handleiding voor de Child Behavior Checklist (CBCL)[END_REF] and the Teacher Report Form (TRF) [START_REF] Achenbach | Teacher Report Form (TRF)[END_REF][START_REF] Verhulst | Handleiding voor de Teacher Report Form (TRF)[END_REF].

Procedure

This study was approved by the Ethical Committee of Ghent University Hospital, Belgium. Children with ADHD were recruited from the child psychiatric outpatient unit of the University Hospital Ghent (Belgium).

After parents of children with ADHD provided written consent, the school and teacher of the child with ADHD were contacted to seek their collaboration in the study. Teachers primarily selected three sex-and age-matched normally developing classmates without formal psychiatric or medical diagnosis. If all parents of the control group provided written consent, one control child was selected randomly to participate in the study.

Observations were carried out in the naturalistic home and school environment of the children during five consecutive days (three school days and two weekend days). To random the order of weekdays and weekend days, 16 paired (ADHD and control classmate) observations were randomly carried out from Tuesday evening until Sunday evening, and 14 paired observations from Friday evening until Wednesday evening. During schooldays, the child with ADHD and its matched control were exposed to similar contextual conditions. School started between 8:20 and 8:45 AM and finished between 3:30 and 4:00 PM, with recess/lunch periods between 10:00 and 10:30 AM, between noon and 1:30 PM, and between 2:30 and 3:00 PM. In general, academic lessons (mathematics, language arts) were mostly planned in the morning (70 % vs 30 % in the afternoon), whereas nonacademic lessons (music and arts) were mostly scheduled in the afternoon (90 % vs 10 % in the morning).

On the start day of the study, all persons involved attended a meeting during which the course of the study was briefly recapitulated. Length, weight, and Tanner pubertal stage of all children were assessed. Parents reported on children's caffeine use and the presence of nightmares. A detailed explanation on activity and heart rate devices was given. Parents were instructed how children should wear the Actiwatch and practiced how to attach the Actiheart and replace monitoring electrodes. Children received a small reward for their participation.

Measures

Demographic variables

Because of their possible influence on heart rate levels, age, sex, Tanner pubertal stage and body BMI were taken into account [5,[START_REF] Wallis | Age related reference ranges for respiration rate and heart rate from 4 to 16 years[END_REF]. Additionally, average caffeine use and the presences of nightmares in all children was questioned.

CBCL/TRF

The CBCL [START_REF] Achenbach | Child Behavior Checklist for ages 4-18 (CBCL/4-18)[END_REF][START_REF] Verhulst | Handleiding voor de Child Behavior Checklist (CBCL)[END_REF] screens for emotional and behavioral problems in children aged 4 to 18. The CBCL was completed by parents, and a parallel version, the TRF [START_REF] Achenbach | Teacher Report Form (TRF)[END_REF][START_REF] Verhulst | Handleiding voor de Teacher Report Form (TRF)[END_REF], was completed by classroom teachers of all children. The CBCL and TRF contain items rated on a 3-point Likert scale yielding three broadband scales: internalizing, externalizing, and total problems. Both questionnaires have good reliability and validity.

The prevalence of comorbid psychiatric problems was high in our sample of children with ADHD. As both externalizing problems (such as ODD and CD) and internalizing problems (such as anxiety and depression) have been related to lower [START_REF] Dietrich | Externalizing and internalizing problems in relation to autonomic function: A population-based study in preadolescents[END_REF][START_REF] Lorber | Psychophysiology of aggression, psychopathy, and conduct problems: A meta-analysis[END_REF][START_REF] Mezzacappa | Anxiety, antisocial behavior, and heart rate regulation in adolescent males[END_REF][START_REF] Ortiz | Heart rate level and antisocial behavior in children and adolescents: A meta-analysis[END_REF] and higher [START_REF] Dietrich | Externalizing and internalizing problems in relation to autonomic function: A population-based study in preadolescents[END_REF][START_REF] Mezzacappa | Anxiety, antisocial behavior, and heart rate regulation in adolescent males[END_REF] heart rate levels respectively, the possible influence of externalizing and internalizing problems (as measured by the CBCL) on heart rate levels was taken into account.

Actiheart: 24-hour heart rate patterns

Heart rate was measured with the Actiheart (Cambridge Neurotechnology Ltd, Cambridge, UK), a small single-piece device with a combined heart rate and movement sensor, designed to clip on to two standard ECG electrodes (3M Ag/AgCl 2223, Germany), positioned at the level of the third intercostal space [START_REF] Crouter | Accuracy of the Actiheart for the assessment of energy expenditure in adults[END_REF]. More detailed technical information on this device has been reported by Brage et al. [START_REF] Brage | Reliability and validity of the combined heart rate and movement sensor Actiheart[END_REF].

In this circadian study, the epoch length was set to 1 min to make longer term evaluation possible.

Through an interface, the data were loaded into a software program (Actiheart software, version 2.4, Cambridge Neurotechnology Ltd, Cambridge, UK) and cleaned before further analysis so that bathing and swimming periods were excluded from the analysis. For data analysis, heart rate counts were averaged per hour.

Actigraph: 24-hour activity patterns

An actigraph is an acceleration-sensitive, watch-like device typically worn on the non-dominant wrist (Actiwatch, Cambridge Neurotechnology Ltd, Cambridge, United Kingdom). For the purpose of this study, an epoch length of 1 min was set. Each time an actigraph is moved compared to a reference signal a voltage is generated. The number of movements during a 1 min interval is accumulated and saved before resetting the counter to zero, providing a measure of activity frequency (counts per minute). Through an interface, the actigraph data were loaded into a software program (Sleep Analysis, version 7, Cambridge Neurotechnology Ltd, Cambridge, UK) for further analysis. For the purpose of this study, actigraph data were evaluated in terms of absolute intensity by calculating activity counts per hour.

Statistical analyses

Demographic variables (age, sex, BMI, pubertal stage, caffeine usus, and presence of nightmares) and scores on the CBCL and TRF were compared between ADHD and control group using analysis of variance (ANOVA) or chi-square tests as appropriate.

Mixed model analysis was conducted with SAS PROC MIXED (SAS, version 9.2) to test whether 24hour heart rate and activity data differed between ADHD and control group. This method is preferred above repeated measures ANOVA for the study of longitudinal data [START_REF] Bagiella | Mixed-effects models in psychophysiology[END_REF][START_REF] Gueorguieva | Move over ANOVA -Progress in analyzing repeatedmeasures data and its reflection in papers published in the archives of general psychiatry[END_REF]. Likelihood ratio tests were used to define the best fitting mixed model for heart rate and activity separately. Different plausible covariance-pattern models, namely heterogeneous compound symmetry, heterogeneous autoregressive, and unstructured covariance patterns were fitted, with or without inclusion of random intercept effects for subject and for day of evaluation within subject in the model. Main predictors of heart rate and activity were diagnostic group (ADHD versus control), time of day (24 hourly data points), day of evaluation (day one to five), and weekday versus weekend day. An interaction term was included for group x time of day to detect differential group effects across time. The heart rate model statement was expanded with other possible confounders of heart rate levels. In the first step, demographic variables (age, sex, pubertal stage, BMI, caffeine use, nightmares) were entered. In the second step, comorbid psychiatric problems (internalizing, externalizing) were entered. Predictors that did not significantly contribute to variations in heart rate were stepwise excluded from the model (significance level at 0.10).

In order to interpret effects obtained with the mixed model, several contrast analyses were additionally performed. First, polynomial time-related contrasts (i.e., linear, quadratic, and cubic effects) were defined for time and for group x time effects. Second, groups were contrasted to evaluate differences in heart rate and activity levels at each time point. To help interpret clinical relevance of statistically significant effects, effect sizes (ES) were calculated as the standardized mean difference between groups (Cohen delta or d = M 1 -M 2 / SD pooled ), which is independent of sample size. ES are defined as small (0.2), moderate (0.5), and large (0.8).

Finally, post hoc analyses were performed to address time-to-time (i.e., peak-to-through) changes across ADHD and control individuals during school days, when contextual influences were similar for both groups. ES were calculated as partial eta squared (η p 2 ) to take into accounted associations between repeated measurement data (time x vs time x+1 ). ES are defined as small (0.01), moderate (0.06), and large (0.14). Comparison of the school versus home assessment was used to explore whether time-to-time effects could be related to environmental school influences.

RESULTS

Demographic variables and current emotional and behavioral problems

Age, sex, BMI, pubertal stage, caffeine use and presence of nightmares did not differ between groups (see Table 1). Children with ADHD showed higher ratings on all scales of CBCL and TRF -except for somatic complaints-compared to normal controls (see Table 1).

INSERT TABLE 1 ABOUT HERE

24-hour heart rate patterns

Considering heart rate, a model with an unstructured covariance pattern including a random intercept for subject fitted best the data. This model revealed a significant effect of diagnostic group (F(1,7030) = 7.81; p < 0.01), time of day (F(23,7030) = 135.54; p < 0.0001), but no effect of diagnostic group x time of day (F(23,7030) < 1; ns), in a model controlling for day of evaluation (F(4,7030) = 2.61; p < 0.05), weekday versus weekend day (F(1,7030) < 1; ns), and age (F(1,7030) = 11.40; p < 0.001). Sex, BMI, pubertal stage, caffeine use, presence of nightmares, and comorbid externalizing and internalizing problems were excluded from the final model as these factors did not significantly contribute to variations in heart rate levels.

Polynomial contrasts of the time effect showed a large significant quadratic effect (t(7030) = -43.36; p < 0.001), next to (smaller) significant linear (t(7030) = 17.45; p < 0.001) and cubic (t(7030) = -18.43; p < 0.001) effects. These effects were reflected in the solutions of fixed effects; heart rate levels were significantly higher during daytime than during nighttime in both groups, with an increase in heart rate levels from early morning to noon (steep from 6 AM to 8 AM and more gradual to noon) and a gradual decrease from noon to evening and night.

Overall, heart rate levels were significantly higher in the ADHD group compared to the normal control group. However, polynomial contrasts of the group x time effect showed a significant cubic effect (t(7030) = 2.33; p < 0.05), reflecting that group differences were particularly expressed at specific times of the day.

Contrast analyses confirmed that children with ADHD showed higher heart rate levels as compared to controls, but that these effects were larger during nighttime and afternoon (moderate ES) than during morning hours (small ES) (see Table 2). A visual representation of group differences in heart rate levels, averaged across 5 days, is presented in Figure 1.

INSERT TABLE 2 ABOUT HERE INSERT FIGURE 1 ABOUT HERE

24-hour activity patterns

Considering activity, a model with a heterogeneous autoregressive symmetry pattern and a random intercept for subject and for day within subject effect fitted the data best. This model showed a significant effect of diagnostic group (F(1,6854) = 4.33; p < 0.05), time of day (F(23,6854) = 148.23; p < 0.0001) and a significant interaction effect of diagnostic group x time of day (F(23,6854); p < 0.05), in a model controlling for day of evaluation (F(4,235) < 1; ns) and weekday versus weekend days (F(1,6854) < 1; ns).

Polynomial contrasts of the time effect showed a large significant quadratic effect (t(6854) = -45.88; p < 0.001), next to (smaller) significant linear (t(6854) = 12.66; p < 0.001) and cubic (t(6854) = -11.91; p < 0.001) effects. These effects were reflected in the solutions of fixed effects; activity levels were significantly lower during night than during day hours in both groups, with an increase in activity levels from early morning to noon and a decrease from noon to nighttime levels.

Polynomial contrasts of the group x time effect showed a significant quadratic effect (t(6854) = 2.83; p < 0.01), reflecting group differences at specific times of the day, but not during others. During day hours, more specifically between 8 and 9 AM and between 11 AM and 4 PM, activity levels were significantly higher in the ADHD compared to the normal control group, but no group differences were observed during evening and night hours. Contrast analyses confirmed moderate to large group effects between 11 AM and 4 PM, small to moderate effects in the morning, but no evidence for group differences during evening and night (see Table 2). A visual representation of group differences in activity levels, averaged across 5 days, is presented in Figure 1.

Possible contextual influence on diurnal variations in activity and heart rate

Mixed model analyses described above did not reveal evidence for significant different effects of weekday and weekend days on activity and heart rate variations. This was confirmed by analyses for weekday and weekend days separately yielding results comparable to our pervious findings (polynomial effects for group x time: heart rate (weekday: p < 0.05; weekend day: p < 0.05) and activity (weekday: p < 0.01; weekend day: p < 0.01)). A visual representation of group differences in heart rate and activity levels, averaged across weekdays and weekend days separately, is presented in Figure 2.

INSERT FIGURE 2 ABOUT HERE However, variations in ES comparing groups at each time point across weekday and weekend days suggest that more subtle weekday/weekend effects may be present (see Table 2); for example, large group effects at 9 AM on weekdays but not on weekend days. Compared to the overall large time of day effect, small effects produced by school schedules (i.e., classroom and playground transitions) may be superimposed on the overall activity and heart rate pattern of children with and without ADHD. Therefore, contrast analyses were performed to estimate effect size of the peak-to-through differences during school hours (i.e., between 8 AM and 4 PM) in both groups (see Table 3). In general, subsequent time-to-time changes in activity were very large between 8 AM and 1 PM for both groups. Only for 10 to 11 AM, children with ADHD showed no significant drop in activity levels in their transition from playtime to class condition. Time-to-time changes in heart rate reflect similar patterns as described for activity except for the noon to 2 PM change. Similar analyses for weekend days could not reveal these large time-to-time changes in the morning (see Table 3). In the afternoon, both groups showed smaller time-to-time variations in activity and heart rate during weekdays and weekend days.

INSERT TABLE 3 ABOUT HERE

DISCUSSION

To the best of our knowledge this is the first study investigating alterations in diurnal arousal patterns in children with ADHD by 24-hour registration of heart rate combined with activity.

The main aim of this study was to assess whether 24-hour heart rate patterns in non-medicated children with ADHD, registered during five days, under natural environmental conditions, were different as compared to normal controls. Although a typical circadian heart rate pattern with low levels during night and high levels during day hours was revealed [START_REF] Scheer | Cardiovascular control by the suprachiasmatic nucleus: Neural and neuroendocrine mechanisms in human and rat[END_REF][START_REF] Waterhouse | Chronophysiology of the cardiovascular system[END_REF], we found no significant interaction effect of time of day x group. Heart rate levels in children with ADHD were overall significantly higher than in control children, independent of comorbid internalizing and externalizing problems. However, additional contrast analyses revealed that these group effects were larger at specific times of day, namely afternoon and nighttime hours. Especially the increase in nighttime levels, i.e., resting heart rate, seems important as this measure refers to a lower vagal tone [START_REF] Thayer | Psychosomatics and psychopathology: looking up and down from the brain[END_REF][START_REF] Thayer | The role of vagal function in the risk for cardiovascular disease and mortality[END_REF].

Our results are in line with Tonhajzerova et al. [START_REF] Tonhajzerova | Changes in the cardiac autonomic regulation in children with attention deficit hyperactivity disorder (ADHD)[END_REF], who found during their short-term evaluation that nonmedicated children with ADHD had tachycardia compared to normal controls in both supine position and orthostasis. These results reflected an autonomic imbalance -low parasympathetic activity and a relative sympathetic dominance-, indicated by low heart rate variability, which the authors hypothesized to be a marker for prefrontal hypo-activity. Thayer and Sternberg [START_REF] Thayer | Beyond heart rate variability -Vagal regulation of allostatic systems[END_REF] described the prefrontal cortex to be part of the central autonomic network (CAN), an entity that controls both visceromotor and behavioral responses that are critical for goal-directed behavior and adaptability. As such they suggested that deficits in the prefrontal cortex may lead to a relative sympathetic dominance as the frontal cortex normally exerts tonic inhibition on brainstem sympathoexcitatory circuits. In ADHD, such a prefrontal hypofunction has been shown with respect to structural and functional neuroimaging, dopaminergic and noradrenergic activity, and neuropsychological executive functioning [START_REF] Arnsten | Catecholamine modulation of prefrontal cortical cognitive function[END_REF][START_REF] Halperin | Revisiting the role of the prefrontal cortex in the pathophysiology of attention-deficit/hyperactivity disorder[END_REF]26,[START_REF] Valera | Meta-analysis of structural imaging findings in attention-deficit/hyperactivity disorder[END_REF].

Additionally, this study simultaneously registered activity to adjust for locomotor-driven increases in heart rate. Analyses showed a diurnal pattern with low levels during night and high levels during day hours (especially afternoon) in line with previous reports [START_REF] Deforche | Objectively measured physical activity, physical activity related personality and body mass index in 6-to 10-yr-old children: a cross-sectional study[END_REF][START_REF] Riddoch | Objective measurement of levels and patterns of physical activity[END_REF]. This diurnal pattern seems to be influenced by environmental conditions and activities. For example, both groups showed a significant peak at 10AM and at 12 AM during school days -but not weekend days-reflecting recess and lunchtime periods. This finding suggests that school-related context factors (i.e., classroom and playground transitions) may be responsible for peak-tothrough variations in activity levels, especially in the morning. In the afternoon, activity levels in both groups were higher than at 9 AM and at 11 AM class conditions. Though this could be due to the planning of more academic lessons in the morning and less academic lessons in the afternoon, a similar morning-afternoon effect was observed during weekend days. This finding indicates that morning to afternoon variations in activity are -at least partly-independent of class schedules in children with and without ADHD. However, a significant interaction effect of diagnostic group x time of day was observed. Compared to control children, children with ADHD had higher activity levels during daytime, especially during noon and early afternoon hours, but surprisingly, not during night. Boonstra et al. [START_REF] Boonstra | Hyperactive night and day? Actigraphy studies in adult ADHD: a baseline comparison and the effect of methylphenidate[END_REF] described a similar pattern in adults with ADHD, i.e., high but stable activity levels during day hours, but no evidence of significant difference at night. Daytime activity levels in children with ADHD may be more variable as differential effects between children with ADHD and control children were described to be particularly expressed when a highly demanding environment and afternoon hours overlap [START_REF] Porrino | A naturalistic assessment of the motor-activity of hyperactive boys. 1. Comparison with normal controls[END_REF]60]. In the morning, children with ADHD showed no significant drop in activity from the 10 AM playtime to the 11 AM class condition compared to their normal classmates. Such settling-down problems after recess periods in ADHD have previously been described [3]. As this study did not systematically assess environmental factors, it is not sure whether the interaction effect in activity levels refers to a true 'circadian' alteration. Especially the fact that no group difference was observed during nighttime in this study could point towards an important role of environmental conditions to express arousal-related fluctuations in ADHD behavior.

In addition, group differences were only detected between 8 AM and 4 PM, indicating the possible involvement of school tasks that were similar for each ADHD and control dyad during this period. Alternatively, it is unlikely that only environmental factors would be responsible for these differences as weekday and weekend days revealed comparable findings considering the expression of ADHD hyperactivity.

In relation to heart rate, we hypothesized that elevated heart rate levels could be driven by higher locomotor activity in the ADHD group. Even though this could be true considering day hours, no such confounding effect of activity on heart rate levels could be retained during night. Nighttime activity levels were not significantly different between groups and could therefore not be responsible for higher nighttime heart rate levels. Although we expected a close relation between heart rate and activity levels, statistically, more 'continue' problems with the inhibition of sympathetic activity did not coincide with the 'specific time of day' problems with the inhibition of behavior. In his polyvagal theory, Porges [START_REF] Porges | Orienting in a defensive world: Mammalian modifications of our evolutionary heritage. A polyvagal theory[END_REF] described two vagal systems: one system associated with the regulation of visceral functions (e.g., heart rate), the other with processes of attention, motor activity, emotion, and social engagement. Although heart rate and activity vary constantly to adapt to changing environmental demands [START_REF] Thayer | The role of vagal function in the risk for cardiovascular disease and mortality[END_REF], these two systems employ different adaptation strategies [START_REF] Porges | Orienting in a defensive world: Mammalian modifications of our evolutionary heritage. A polyvagal theory[END_REF].

There are a number of issues to consider in evaluating this research as besides activity, also other factors could have elevated heart rate levels in ADHD. First, higher stress levels are expected to elevate heart rate levels so that differential results in ADHD may be due to secondary conditions rather than to ADHD itself.

Next to the prefrontal cortex, also limbic structures are part of the CAN and a close association between these regions and baroreflex functions has been described [START_REF] Thayer | Psychosomatics and psychopathology: looking up and down from the brain[END_REF]. It has been suggested that anomalies in heart rate could be an early peripheral marker of latent asymptomatic and undiagnosed comorbid emotional disorders which may be the result of accumulated stress [START_REF] Tonhajzerova | Changes in the cardiac autonomic regulation in children with attention deficit hyperactivity disorder (ADHD)[END_REF]. In our study, we looked at confounding effects of psychiatric comorbid problems as physiological dysregulation of the autonomic nervous system has been related to psychiatric disorders other than ADHD [START_REF] Dietrich | Externalizing and internalizing problems in relation to autonomic function: A population-based study in preadolescents[END_REF][START_REF] Lorber | Psychophysiology of aggression, psychopathy, and conduct problems: A meta-analysis[END_REF][START_REF] Ortiz | Heart rate level and antisocial behavior in children and adolescents: A meta-analysis[END_REF]. However, we found no evidence that comorbid externalizing and internalizing problems contributed significantly to variations in heart rate levels.

Second, pubertal stage and sex have been reported to influence heart rate levels [5,[START_REF] Wallis | Age related reference ranges for respiration rate and heart rate from 4 to 16 years[END_REF] . Most children in our sample were pre-pubertal without evidence for differences between the ADHD and control group. However, differences for boys and girls in onset of puberty is expected to be about 2 years so that sex differences may be present in our sample but could not be detected due to the small number of girls (n = 6) per group in the study. Additional analyses of boys only (n = 24 per group) revealed similar results to those obtained for the whole sample.

Third, the confounding effect of stimulant medication must be considered as significant increases in heart rate, especially during the first 6 months of treatment, have been reported [START_REF] Daniels | Cardiovascular effects of methylphenidate[END_REF]. Although all participants were free from medication at least 72 hours prior to participation, to date, it is not sure whether fast-release preparations of methylphenidate could still have influenced cardiovascular functioning. In this study, day-to-day variability in heart rate levels however showed no decrease in the effect pattern from day one to day five.

Additional analyses showed that midnight heart rate levels were significantly higher on day 5 than on day one (p < 0.01), reducing the possibility that our findings can be explained by longer-term effects of stimulant medication.

Fourth, confounders such as caffeine use and the presence of nightmares may have influenced day-and nighttime heart rate levels. These factors however were not significantly different between groups and did not significantly contribute to heart rate variations. None of the children in our sample used nicotine, alcohol, or illicit drugs.

Finally, regular physical training is known to lower resting heart rate levels. However, we did not systematically assess the amount of sport activities and outdoor play in our sample. Therefore, we cannot exclude the possibility that such activities could have affected heart rate in our control more than in our ADHD group.

In the light of these considerations, our finding of elevated heart rate levels, i.e., hyperarousal, may suggest the necessity to critically re-evaluate the arousal concept in ADHD. Theoretical frameworks [START_REF] Sergeant | Convergence of approaches in localizing the hyperactivity deficit[END_REF][START_REF] Zentall | Optimal stimulation -a model of disordered activity and performance in normal and deviant children[END_REF] have mostly defined ADHD in terms of hypo-arousal deficits -largely ignoring circadian changes in arousal. In our study, we found time of day effects to be important as higher heart rate levels in ADHD were particularly expressed during night and afternoon hours. To our knowledge, nighttime heart rate levels have not been assessed yet in children with ADHD. During daytime periods however, low resting heart rate has consistently been reported in aggressive and antisocial behavior [START_REF] Lorber | Psychophysiology of aggression, psychopathy, and conduct problems: A meta-analysis[END_REF][START_REF] Ortiz | Heart rate level and antisocial behavior in children and adolescents: A meta-analysis[END_REF]. This finding may not blindly be generalized to ADHD as studies in this population revealed inconsistent results with most of them describing no difference in resting heart rate [START_REF] Iaboni | Psychophysiological response of ADHD children to reward and extinction[END_REF][START_REF] Lackschewitz | Physiological and psychological stress responses in adults with attention-deficit/hyperactivity disorder (ADHD)[END_REF]. Next to possible confounding effects of time of day, these baseline levels may be influenced by varying levels of anticipatory stress as 'resting' assessments were mostly carried out prior to a laboratory experiment investigating the environmental effects on arousal. Environmental influences may be superimposed on the diurnal pattern as was suggested by more variable effect sizes during daytime in our study.

A wealth of studies in subjects with ADHD reported hypo-aroused heart rate responses in relation to for example stress [START_REF] Lackschewitz | Physiological and psychological stress responses in adults with attention-deficit/hyperactivity disorder (ADHD)[END_REF] and reward [START_REF] Crone | Sensitivity to interference and response contingencies in attention-deficit/hyperactivity disorder[END_REF][START_REF] Iaboni | Psychophysiological response of ADHD children to reward and extinction[END_REF]. It is difficult however to compare these findings to those of our study as we did not investigate heart rate in response to such stimuli, under strict laboratory conditions. Nevertheless, we carefully explain our findings in the light of hypo-aroused responsiveness in ADHD. First, it seems that smaller group differences in morning hours may reflect a deviation from the overall higher arousal level, i.e., 'relative hypo-arousal', in children with ADHD during moments of higher academic demands. This effect was however also present throughout weekend days during which cognitive involvement is less plausible. Second, in relation to activity, blunted heart rate responses were reported during peak exercise in ADHD [START_REF] Mahon | Exercise Responses in Boys With Attention Deficit/Hyperactivity Disorder Effects of Stimulant Medication[END_REF]. For a larger min-max difference in activity in children with ADHD (i.e., 812 counts/min for ADHD and 694 for control; p < 0.01), our findings reveal a comparable min-max difference in heart rate (i.e., 40 bpm for ADHD and 39 for control; ns), possibly indicating a lower dynamic autonomic potential in children with ADHD than could be expected based on their activity pattern. Alternatively, hyperaroused response patterns have been proposed in ADHD as sustained by higher heart rate levels in response to for example reward [START_REF] Luman | Decision-making in ADHD: sensitive to frequency but blind to the magnitude of penalty?[END_REF]. Hyperarousal deficits have additionally been supported by the use of α 2 adrenergic agonists such as clonidine for the effective treatment of ADHD symptoms [START_REF] Scahill | Alpha-2 Adrenergic Agonists in Children with Inattention, Hyperactivity and Impulsiveness[END_REF]. These drugs, originally developed for the treatment of hypertension, decrease the firing of noradrenalin neurons in the locus coeruleus, resulting in decreased arousal [START_REF] Arnsten | The contribution of alpha 2-noradrenergic mechanisms of prefrontal cortical cognitive function. Potential significance for attentiondeficit hyperactivity disorder[END_REF]. A similar action of atomoxetine, a noradrenalin re-uptake inhibitor used to treat ADHD, has been described. Li et al. [START_REF] Li | Atomoxetine changes rat's HR response to stress from tachycardia to bradycardia via alterations in autonomic function[END_REF] found an autonomic shift from tachycardia to bradycardia (decrease of sympathetic and increase of parasympathetic tone) after drug administration in rats. This contrasts with the action of methylphenidate, a dopaminergic agent, that is known to increase arousal states in ADHD. Altogether, these findings point to the involvement of time-and context-related aspects in the dynamic modulation of arousal deficits in ADHD, which should be taken into account in future research evaluating deviations from 'optimal' arousal levels (hypoand hyperarousal) in ADHD.

In summary, we suggest an autonomic imbalance with overall higher heart rate levelsparticularly expressed during night and afternoon hours-in our sample of non-medicated children with ADHD. This finding seems independent from activity levels or comorbid psychiatric problems. However, further research is needed to validate our results in a larger sample with heart rate variability as a measure for autonomic function and to assess longer-term effects of these findings in children with ADHD. Higher resting heart rate may have a major impact on health outcomes as an association with higher morbidity and mortality from both cardiovascular and non-cardiovascular diseases has been described [START_REF] Greenland | Resting heart rate is a risk factor for cardiovascular and noncardiovascular mortality -The Chicago Heart Association Detection Project in Industry[END_REF][START_REF] Reunanen | Heart rate and mortality[END_REF]. Also the long-term use of psychostimulants must be questioned if overall heart rate levels are 'truly' increased in children with ADHD.

Figure 1

 1 Figure 1 24-hour heart rate (top) and activity (bottom) pattern in children with ADHD and controls, averaged over five days

Table 1

 1 Heart rate ActivityM diff SD pooled ES (weekdays/weekend days) M diff SD pooled ES (weekdays/weekend days)Note. M diff = mean difference between groups; SD pooled = pooled standard deviation; ES = effect size (Cohen d) ***p < 0.001; **p < 0.01; *p < 0.05; °p < 0.10 Test statistics of demographic variables and total problem ratings on the CBCL and the TRF

	11 PM 3.49 8.28	0.42 (0.81**/0.25)	17.83	84.78	0.21 (0.40/0.23)
			ADHD	Control
			M (SD) / n (%)	M (SD) / n (%)	F(1,59) / χ 2 (1)
	Age		8.97 (1.50)	8.80 (1.54)	F(1,59) < 1, ns
	Sex (n)		24 m (80), 6 f (20)	24 m (80), 6 f (20)	χ 2 (1) < 1, ns
	Pubertal stage
	-Pre-pubertal pubic hair stage (n)	23 m (96), 5 f (83)	24 m (100), 5 f (83)	χ 2 (1) < 1, ns
	-Pre-pubertal genital stage (n)	24 m (100)	24 m (100)	χ 2 (1) < 1, ns
	-Pre-pubertal breast stage (n)	5 f (83)	5 f (83)		χ 2 (1) < 1, ns
	BMI		16.12 (1.54)	15.88 (1.62)	F(1,59) < 1, ns
	Caffeine usus (n)		10 (33)	11 (37)		χ 2 (1) < 1, ns
	Presence of nightmares (n)		3 (10)	1 (3)		χ 2 (1) = 1.41, ns
	CBCL internalizing		62.77 (8.24)	46.23 (9.25)	53.48***
	CBCL externalizing		66.63 (7.70)	44.67 (8.52)	109.74***
	CBCL total		68.90 (6.40)	42.60 (9.54)	156.82***
	TRF internalizing		52.80 (6.62)	46.10 (6.31)	16.12***
	TRF externalizing		58.87 (9.10)	46.20 (5.34)	43.21***
	TRF total		59.63 (5.96)	42.70 (5.90)	122.33***

Acknowledgements

We thank Jan Van Proosdij and Tom Evans for their technical support on actiwatch and actiheart devices, and Dr. Stijn Vansteelandt for his critical comments on statistical analyses. This work was financially supported by the Fund for Scientific Research, Flanders, Belgium (3G032607).

Financial disclosures

Lindita imeraj, Inge Antrop, Ellen Deschepper, Sarah Bal, and Dirk Deboutte report no financial interests or potential conflicts of interest. Herbert Roeyers has served as an advisor to Shire and received research support from Shire and Lilly and conference attendance support from Lilly. The present study is unrelated to these relationships.

Note. ***p < 0.001, **p < 0.01, *p < 0.05, °p < 0.10


K K Wijesundara

C Negulescu

E Foerster

D Monfort

Identification of modal parameters of ambient excitation structures structures using continuous wavelet transform

Keywords: measures, continuous wavelet transform, ambient vibration

Continuous wavelet transform (CWT) has recently emerged as a promising tool for identification of modal properties such as natural frequencies, damping ratios and mode shapes through ambient excitation measurements of structures. This paper mainly discusses the capability of CWT method to identify the modal properties accurately using a numerical application in five storeys frame and two practical applications in a five storeys reinforced concrete structure with masonry in-fill walls and a three storeys masonry structure. Natural frequencies are identified by extracting windows paralleled to the frequency axis at wavelet ridges. Damping coefficients are estimated using the wavelet-based logarithmic decrement method. Finally, a conclusion can be drawn that the continuous wavelet transform used for the output-only system identification of ambient excitation structures yields a good agreement with results of the random decrement method and the finite element models.

INTRODUCTION

Currently, ambient vibration measurements are commonly used in the assessment and structural health monitoring of civil engineering structures such as buildings, bridges and towers because the ambient vibration testing is cheap and fast, no elaborate excitation equipment are required, no boundary condition simulations are required and modal properties of the whole system can be estimated using a modal extraction technique. Furthermore, the extracted modal parameters such as natural periods, mode shapes and damping ratios can also be used for verifying the design characteristics of a civil engineering structure and validating the numerical model that can be used to predict the response of a structure under an extreme loading condition. Many modal extraction techniques found in the literature, which are developed to estimate the modal properties using only the output response, can be categorized into three main groups as time domain, frequency domain and time-frequency domain methods. The frequency domain methods such as frequency domain decomposition (FDD) (Brincker et al. (2001)) and enhanced frequency domain decomposition (EFDD) (Brincker et al. (2001)) are commonly used for the extraction of natural frequencies and mode shapes, but uncertainty in the estimation of damping ratios. In general, time domain methods such as random decrement (RD) method [START_REF] Cole | On-line Failure Detection and Damping Measurements of Aerospace Structures by Random Decrement Signature‖[END_REF]), Ibrahim time domain (ITD) [START_REF] Ibrahim | Random Decrement Technique for Modal Identification of Structures[END_REF]) and the eigen-system realization algorithm (ERA) [START_REF] Juang | An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction‖[END_REF]) are widely used for damping estimation. The main problem associated with many time domain methods is how to distinguish structural modes from uncorrelated modes, which are introduced to accommodate measurement noise, leakage, out-of-band modes and non-linearity, etc. This problem can cause severe error in damping estimation. Recently, continuous wavelet transform (CWT) method is used for modal identification of civil engineering structures using only the output response as a time-frequency domain method. As the CWT method can decompose of a signal into time-frequency domain using a mother wavelet, multi degree of freedom (MDOF) systems can be handled directly. Furthermore, it can work as a band-pass filter and hence, this method can handle very noisy measurements [START_REF] Staszewski | Identification of Damping in M.d.o.f. Systems using Time-scale Decomposition[END_REF]). Another advantage of the WT method is that the stationary assumption for an ambient vibration response is not required. Therefore, this method has many advantages over the other methods in identifying modal properties using ambient vibration measurements. This method has been used successfully to extract the natural frequencies and the associated mode shapes using ambient vibration measurements of a structure (Meo et al. (2006), Le and Tamure (2009)). Regarding the damping estimation, past studies by [START_REF] Staszewski | Identification of Damping in M.d.o.f. Systems using Time-scale Decomposition[END_REF], [START_REF] Hans | Damping identification in multi-degree-of-freedom systems via a wavelet-logarithmic decrement-part 2: study of a civil engineering building‖[END_REF], Lamarqre et al. (2000) and [START_REF] Ta | Identification of Weak Nonlinearities on Damping and Stiffness by the Continuous Wavelet Transform‖[END_REF] have highlighted that damping ratios can be estimated adequately accurate through wavelet-based logarithmic decrement for lightly damped system. However, their studies are limited to either the impulse response or free decay response of structures. The main objective of this study is to obtain the modal properties through ambient vibration measurements of low-rise buildings using CWT method.

CONTINUOUS WAVELET ANALYSIS

This section introduces a brief description on the theoretical background of the wavelet transform. However, authors strongly recommend to readers to refer the key papers to understand the theoretical background of the method well. Fourier transformation, which is a linear transformation, transform a given function x(t) in the time domain into the frequency domain using a basic function e-jwt . This can be represented in the following form:

dt e t x FT jwt x (1)
This transformation does not include any local time information of the function x(t). Initially, Fourier transformation of the short time sliding window is used to overcome this limitation by decomposing of the function into frequency-time domain. However, this method called Short Time Fourier Transformation (STFT) suffers from time-frequency resolution limitation. Later, CWT is developed to obtain better spectral decomposition as an alternative method to STFT. The basic idea of the CWT is to find a function ψ(t), which can generate a basic for the entire domain of function x(t), if a function x(t) satisfies the condition that x(t) decays to zero at +-α as in the case of Fourier transform. The fast decay in time domain and the limited band width in frequency domain introduces locality into the analysis, which is not the case of Fourier transform where a global representation can only be obtained. The function ψ(t) that is called as an wavelet must satisfy the two admissibility condition:

(1) ψ(t) must be absolutely integral and square integral:

dt t dt t 2 (2)
(2) ψ(t) must be band limited and has zero mean:

d (3)
Then, using a wavelet, CWT can be used to decompose of a function x(t) into frequency-time domain as defined in the following form:

dt a b t t x a W b a * ) , ( ) ( 1 (4)
where ψ*(t) and b are the complex conjugate of ψ(t) and the parameter localizing the wavelet function in the time domain, respectively and W(a,b) are the CWT coefficients that represent the measure of the similitude between the function x(t) and the wavelet at the time b and the scale a. The complex Morlet wavelet is commonly used for continuous wave transform as a basic function. The complex Morlet wavelet and its Fourier transform can be expressed as:

2 2 2 2 1 t t if e e t c (5) )) ( 2 ( 2 2 1 c f af e af  (6)
f and fc are Fourier frequency and central wavelet frequency. The conversion between Fourier frequency and the scale a can be established as:

a f f f c s (7)
where fs and a are the sampling frequency and the scale, respectively.

NUMERICAL APPLICATION IN A FIVE STOREY FRAME

The first objective of this part of the study is to validate the capability of CWT method to estimate natural frequencies and damping ratios accurately by decomposing of a time domain free decay response of a MDOF system into a time-frequency domain. However, if the response is not free decay, then the two steps procedure is used to estimate the damping ratio in this study. This procedure could cause severe error in damping estimation when the response is non-stationary and recorded in a relatively short duration because RD signatures evaluated in the first step of the procedure may not be proportional to the free decay response. Therefore, the second objective is to quantify the error associated in estimation of damping ratio using the two steps procedure when the response is non-stationary and recorded in a relatively short duration. The third objective is to quantify the error associated in estimation of natural frequencies and mode shapes when the response is non-stationary and lasts in a short duration. It must be noted that frequencies are estimated using single step procedure by extracting a window parallel to the frequency axis of time-frequency plot at wavelet ridges as described in section 2. For these purposes, a 2D numerical model of a five storey frame to represent a MDOF system is analysed linear elastically for 50 ground motions, recorded during the past earthquakes. The stiffness proportional damping modal with 5% of critical damping for the 1st mode of vibration is assumed in the numerical modal. From the modal analysis, the identified frequencies of the first five translational modes are 2. 1,8.2,16.1,27.8 and 38.5Hz. Figure 1 illustrates the first five translational mode shapes.

VALIDATING THE CAPABILITY OF CWT METHOD TO ESTIMATE THE MODAL PROPERTIES

To fulfill the first objective of validating the capability of CWT method to estimate modal properties, free decay responses are used because they avoids the effects of uncertainty in evaluation of free decay responses from other types of vibration responses (for an example, evaluation of a free decay response from an ambient vibration response of a structure using RD method). Therefore, the structure is analyzed for additional 10 seconds after each earthquake to obtain free decay responses at each storey level. Then, the free decay responses are decomposed into a time-frequency domain by using CWT with complex Morlet wavelet. In order to identify the natural frequencies, windows parallel to the frequency axis at the wavelet ridges in time-frequency plot are extracted. Figure 2 Furthermore, a window paralleled to the time axis at the wavelet ridge is extracted from the timefrequency plot to estimate the damping ratio of the corresponding mode of vibration. Figure 3(a) and (b) show the extracted envelop of the CWT coefficient modulus at the frequency of 2.1Hz and its semi-logarithmic plot, respectively. Furthermore, Figure 3(c) shows the comparison between the free decay response and the extracted envelop. They are in good agreement proving that WT coefficients are proportional to the instantaneous amplitude of the response. The estimated damping ratio corresponding to the 1st mode is 5.09% which is very close to the assumed damping ratio of 5%. The difference in the estimation is less than 2%. There is the maximum error of 1% in the estimation of first mode frequency while it is 2% in estimation of damping ratio of the first mode vibration.

ERROR IN DAMPING RATIO ESTIMATION USING TWO STEPS PROCEDURE SETUP

Figure 4 summarizes the two step procedure to estimate damping ratio in this study. Displacement response at the first storey level resulting in the input ground motion is obtained and subsequently, the RD signature is evaluated using the RD method as shown in the step 1(b). It should be noted that the level crossing triggering technique, which is commonly used in the literature, is adopted in this study. 214 number of segments are used in evaluation of the RD signature shown in Figure 4. Finally, the estimated damping ratio is 5.93% which is 18.7% higher estimation than the assumed damping ratio of 5% for the first mode. Table 1 presents the earthquake no., duration of the ground motion, number of segments used to evaluate the RD signature, the duration of the RD signature, estimated damping ratio, estimated period, error in estimation of damping ratio and the error period of each earthquake. Figure 5 indicates that the maximum percentage of the error in the estimation of damping ratio is 26%. shows the variation of error in the estimation of 1st mode frequency of the structure for different ground motions selected in this study. It indicates that the maximum error in the estimation of period is 6.7%. This highlights the capability of CWT method with complex Morlet wavelet to estimate the period adequately accurate even using a response which is non-stationary and lasts in relatively short duration. Figure 7 shows the comparison of the first and the second mode shapes extracted using CWT and the modal analysis. They are in very good agreement.

CONCLUSIONS

This study mainly investigates the capability of the continuous wavelet transformation (CWT) method using Morlet wavelet in estimation of modal properties of low-rise buildings. For this purpose, a numerical is used. Based on the results of the application following conclusions can be drawn.

It can be concluded that the CWT method can estimate the modal properties accurately when the free decay response of a MDOF system is analyzed. Furthermore, when a response is non-stationary and lasts in relatively short duration, the CWT method can estimate the periods and mode shapes adequately accurate. However, a very significant scatter in estimation of damping ratio is observed with the maximum error of 26% when the two steps procedure is used. This could be due the fact that in damping estimation, the evaluation of RD signature for such a non-stationary and short duration record does not represent the free decay response of the structure.

Figure 1 .

 1 Figure 1. Mode shapes of the five translational modes

  (a), (b) and (c) show the free decay displacement response at the top storey level after an earthquake, its time-frequency plot and the extracted window parallel to the frequency axis at the wavelet ridge, respectively. It is clear from Figure2(b) and (c) that the peak value of the modulus of CWT coefficients corresponds to the frequency of 2.1Hz which is very close to the 1st mode frequency obtained from the modal analysis. The difference is less than 1%.

Figure 2 .

 2 Figure 2. (a) Free decay response at top story, (b) its time-frequency resolution and (c) the extracted window parallel to frequency axis at the wavelet ridge

Figure 3 .

 3 Figure 3. (a) the extracted envelope parallel to frequency axis at the wavelet ridge, (b) semi-logarithmic plot of the envelope and (c) comparison of free decay response and the wavelet envelop

Figure 4 .

 4 Figure 4. The two step procedure for estimation of damping ratio

Figure 4 .

 4 Figure 4. As shown in Figure4(d), the amplitudes of the first two cycles are not matched well with the wavelet due the edge effect. Therefore, the first point represents the third peak where the amplitude of the response and the real part of the scaled WT coefficient are very close to each other and the second point represents the peak of the last cycle of the response. Finally, the estimated damping ratio is 5.93% which is 18.7% higher estimation than the assumed damping ratio of 5% for the first mode. Table1presents the earthquake no., duration of the ground motion, number of segments used to evaluate the RD signature, the duration of the RD signature, estimated damping ratio, estimated period, error in estimation of damping ratio and the error period of each earthquake. Figure5indicates that the maximum percentage of the error in the estimation of damping ratio is 26%.

Figure 5 .

 5 Figure 5. Variation of (a) damping ratio (b) period

Figure 6 .

 6 Figure 6. (a) The displacement response at the first story level, (b) its time-frequency plot and (c) wavelet ridge

Figure 7 .

 7 Figure 7. Comparison of the mode shapes obtained using numerical model and the CWT

Table 1 .

 1 Details of the properties

AKCNOWLEDGEMENT

This work has been supported by French Research National Agency (ANR) through RiskNat program (project URBASIS n°ANR-09-RISK-009), by European Union through SISPYR Interreg Project (POCTEFA 2007-2013 number 73/08) and by Languedoc-Roussillon Regional Council.