Dr Véronique Boulenger
email: veronique.boulenger@mrc-cbu.cam.ac.uk

Olaf Hauk

Friedemann Pulvermüller

Grasping ideas with the motor system: Semantic somatotopy in idiom comprehension

Keywords:

Single words and sentences referring to bodily actions activate the motor cortex.

However, this semantic grounding of concrete language does not address the critical question whether the sensory-motor system contributes to the processing of abstract meaning and thought. We examined fMRI activation to idioms and literal sentences including arm-and leg-related action words. A common left fronto-temporal network was engaged in sentence reading, with idioms yielding relatively stronger activity in (pre)frontal and middle temporal cortex. Crucially, somatotopic activation along the motor strip, in central and precentral cortex, was elicited by idiomatic and literal sentences, reflecting the body part reference of the words embedded in the sentences. Semantic somatotopy was most pronounced after sentence ending, thus reflecting sentence level processing rather than that of single words. These results indicate that semantic representations grounded in the sensory-motor system play a role in the composition of sentence level meaning, even in the case of idioms.

Semantic information conveyed by language is reflected in the brain response [START_REF] Barsalou | Perceptual symbol systems[END_REF][START_REF] Barsalou | Grounded cognition[END_REF][START_REF] Martin | The representation of object concepts in the brain[END_REF][START_REF] Pulvermüller | Brain reflections of words and their meaning[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF]. Specific brain activation patterns reveal fine-grained differences between semantic categories such as actions and objects [START_REF] Martin | Semantic memory and the brain: structure and processes[END_REF][START_REF] Pulvermüller | Nouns and verbs in the intact brain: evidence from event-related potentials and high-frequency cortical responses[END_REF], tools and animals [START_REF] Beauchamp | Grounding object concepts in perception and action: evidence from fMRI studies of tools[END_REF][START_REF] Martin | Neural correlates of categoryspecific knowledge[END_REF] as well as color and form (Moscoso del Prado Martin et al. 2006;Pulvermüller and Hauk 2006). fMRI activation along the motor strip distinguishes between words and sentences that refer to actions involving the face, arms or legs [START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF]Hauk et al. 2004;Kemmerer et al. 2008;[START_REF] Pulvermüller | Walking or talking? Behavioral and neurophysiological correlates of action verb processing[END_REF]Tettamanti et al. 2005). This "semantic somatotopy" has provided a major argument supporting the idea that semantic mechanisms are grounded in action-perception systems of the brain [START_REF] Barsalou | Grounded cognition[END_REF][START_REF] Glenberg | Language and action: creating sensible combination of ideas[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF]. A range of behavioral, TMS and neuropsychological studies also supports this view [START_REF] Bak | Clinical, imaging and pathological correlates of a hereditary deficit in verb and action processing[END_REF][START_REF] Boulenger | Crosstalk between language processes and overt motor behavior in the first 200 ms of processing[END_REF]Boulenger et al. , 2008aBoulenger et al. , 2008b;;[START_REF] Buccino | Listening to action-related sentences modulates the activity of the motor system: a combined TMS and behavioral study[END_REF][START_REF] Glenberg | Grounding language in action[END_REF][START_REF] Nazir | Language-induced motor perturbations during the execution of a reaching movement[END_REF][START_REF] Neininger | Word-category specific deficits after lesions in the right hemisphere[END_REF]Pulvermüller et al. 2005a;[START_REF] Zwaan | Seeing, acting, understanding: motor resonance in language comprehension[END_REF].

Previous research on the grounding of semantics in action-perception circuits has, however, suffered from a major shortcoming. Only concrete meaning of single words ("kick" vs. "pick") and sentences ("she kicks the ball" vs. "she picks the pen") has appropriately been examined. Here we ask whether semantic somatotopy in the motor system persists during processing of idiomatic sentence meaning (e.g. "she kicks the habit"). In a previous study, [START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF] examined the pattern of brain activation during reading of metaphorical sentences including action words (e.g. "biting off more that you can chew") compared to action-related literal sentences (e.g. "biting the peach"). While their results showed somatotopic activity in the premotor cortex for literal stimuli, they failed to reveal any significant motor activation for metaphorical sentences. This absence of effects may however be explained by methodological issues such as the limited number of stimuli (5 sentences per condition, repeated 8 times each).

The aim of the present study was to more suitably test and clarify whether the motor system comes into play during comprehension of figurative action-related language. If the grounding of semantics in sensory-motor processes is a universal feature of the human cognitive system [START_REF] Glenberg | What memory is for[END_REF][START_REF] Glenberg | Grounding language in action[END_REF][START_REF] Jeannerod | Motor cognition: What action tells the self[END_REF][START_REF] Lakoff | Philosophy In The Flesh: The Embodied Mind and Its Challenge to Western Thought[END_REF], the prediction is that action-perception information should influence semantic brain activation to sentences, even if their meaning is highly abstract.

To test this, we chose to look at idioms that include words referring to actions performed using the arm and leg (e.g. "He grasped the idea" and "He kicked the habit" respectively) and examined fMRI activation in the motor areas related to upper and lower limbs as the dependent variable.

The study also allowed us to address more general issues in cognitive science.

According to compositional theories of semantics [START_REF] Cacciari | The comprehension of idioms[END_REF][START_REF] Davidson | Truth and Meaning[END_REF][START_REF] Titone | On the compositional and noncompositional nature of idiomatic expressions[END_REF], the meaning of abstract sentences is computed from the meaning of words included in these sentences and from combinatorial information. Similar to a semantic grounding perspective [START_REF] Barsalou | Grounded cognition[END_REF][START_REF] Glenberg | What memory is for[END_REF], compositional theories would therefore receive support from semantic somatotopy to idioms that include action-related words. In this case, it would be argued that the meaning of the constituent words influences fMRI activation patterns at the sentence level [START_REF] Gibbs | Idioms and mental imagery: The metaphorical motivation for idiomatic meaning[END_REF]. Alternatively, abstract idioms could be stored separate from their constituent words as whole units and could be retrieved similar to the way long words are accessed in long-term memory [START_REF] Bobrow | On catching on to idiomatic expressions[END_REF][START_REF] Gibbs | Spilling the beans on understanding and memory for idioms in conversation[END_REF]. In such a "lexicalist" approach to idioms, no action word-related semantic activation and therefore no semantic somatotopy should emerge in the fMRI signature of idioms.

In language comprehension, early lexico-semantic processing of single words [START_REF] Pulvermüller | Word processing in the brain as revealed by neurophysiological imaging[END_REF] for a review) may be followed, after a delay, by the understanding of sentence meaning (Barber and Kutas 2007). In support of this view, metabolic activity related to semantic integration at the sentence level was found to be maximal at about 6-8 s after sentence completion [START_REF] Humphries | Time-course of semantic processes during sentence comprehension: An fMRI study[END_REF] or even later [START_REF] Simmons | fMRI evidence for word association and situated simulation in conceptual processing[END_REF]. In the present study, we chose to monitor brain activation in two time-windows that might reflect different temporal steps during comprehension of idioms. Cortical activity was examined at the onset of the critical word of the sentences ("He grasped the IDEA"), which disambiguated the sentences as either idiomatic or literal (early analysis window), and 3 sec after its end (late analysis window). The prediction was that (1) differences in cortical activation between literal and idiomatic sentences would emerge in both time-windows and that (2) semantic contribution of action words would become evident as semantic somatotopy at the sentence level, i.e. in the late time-window.

Materials and Methods

Participants

Eighteen healthy right-handed native English speakers (eight females) participated in the study. They had normal or corrected-to-normal vision and no history of neurological or psychiatric disorder. The mean age of the volunteers was 24.3 years (SD = 6.3). They were paid for their participation. Ethical approval was obtained from the Cambridge Local Research Ethics Committee.

Materials

Seventy-six pairs of idiomatic and literal English sentences were used in this experiment. In each condition, half of the sentences included an arm-related action word (e.g. "John grasped the idea" and "John grasped the object") and the other half contained a leg-related action word (e.g. "Pablo kicked the habit", which means "to stop doing something that is difficult to stop doing", and "Pablo kicked the ball"). Four experimental conditions were thus compared: arm-related idiomatic sentences (n = 38), arm-related literal sentences (n = 38), leg-related idiomatic sentences (n = 38) and legrelated literal sentences (n = 38). Sentence length varied from 3 to 7 words. The critical words of the sentences (e.g. "idea" and "object"), which disambiguated the sentences as either idiomatic or literal, were matched using the CELEX lexical database for relevant psycholinguistic variables, including word frequency, lemma frequency, length in letters, number of syllables, bigram frequency, trigram frequency and number of orthographic neighbors. Arm-and leg-related action words were matched along the same variables (see Table 1). The two types of sentences were also matched for syntactic structure (i.e. only the critical words differed between idiomatic and literal conditions) and cloze probability. The latter parameter was defined as the number of occurrences, on http://www.google.co.uk, of the critical verb phrase of the sentences (e.g. "grasped the idea"; 53917 ±12571 for idioms vs. 37892 ±13856 for literal sentences, p > .05).

Seventy-six baseline stimuli, consisting of strings of meaningless hash-marks matched in length with the sentences (e.g. "## ####### ### ####"), were also constructed (see Hauk et al. 2004a for similar methods). Finally, six literal sentences (different from and not related to the experimental stimuli; e.g. "John opened the door") were used as probe sentences in a simple motor response task.

-Table 1 about here -

Procedure

The main experiment was run in 2 blocks, each block consisting of 117 trials (76 experimental trials, 38 baseline trials and 3 probe trials). Sentences were presented word-by-word, each for 500 ms (Stimulus Onset Asynchrony, SOA = 500 ms), in lowercase letters at the centre of a computer screen. The SOA between critical words of two consecutive sentences was fixed (6.6 s) and the inter-sentence interval (i.e. time interval between the offset of a sentence and the onset of the next sentence), during which a fixation cross remained on the screen, varied between 2.6 s and 5.1 s (mean = 4.04 s, SE = .05; Figure 1). Participants were given the following instructions: "After display of a fixation cross at the centre of the screen, sentences will be presented word by word.

Please read words silently but attentively. Sequences of symbols will also be displayed, please look at them attentively". They were told to attend to the meaning of each sentence and to be prepared to respond to test questions probing their comprehension.

To this end, they had to answer simple yes/no questions about probe sentences, interspersed between critical sentences, by pressing a button on a two-button response box either with their left index or middle finger. For instance, after reading "John opened the door", they had to answer "no" to the question "Did John open the fridge?". Note that subjects did not know which sentences were probes, i.e. they had to expect questions after any sentence. Stimuli were presented in a randomized order by means of E-Prime software and viewed via a back-projection screen located in front of the scanner and a mirror placed on the head coil.

Subsequent to the main experiment, participants were asked to perform a motor localizer task. The localizer scans always followed the sentence experiment to avoid any attentional bias towards action-related aspects of the stimuli. Instructions on which extremity to move (right or left index finger, right or left foot) were presented visually on the computer screen. Instructions remained on the screen for 20 s each and were repeated four times in pseudo-randomized order (see Hauk et al. 2004).

-Figure 1 about here -

Imaging methods

Subjects were scanned in a 3-T Siemens Tim Trio magnetic resonance system using a head coil. Echo-planar imaging (EPI) sequence parameters were TR = 2 s, TE = 30 ms and flip angle = 78°. The functional images consisted of 32 slices covering the whole brain (slice thickness 3 mm, interslice distance 0.75 mm, in-plane resolution 3 x 3 mm).

Imaging data were processed using SPM5 software (Wellcome Department of Cognitive Neurology, London UK).

Images were corrected for slice timing and then realigned to the first image using sinc interpolation. Any non-brain parts were removed from the T1-weighted structural images by using a surface-model approach (''skull-stripping"; Smith 2002). The EPI images were coregistered to these skull-stripped structural T1 images by using a mutual information coregistration procedure [START_REF] Maes | Multimodality image registration by maximization of mutual information[END_REF]). The structural MRI was normalized to the 152-subject T1 template of the Montreal Neurological Institute (MNI).

The resulting transformation parameters were applied to the coregistered EPI images.

During the spatial normalization process, images were resampled with a spatial resolution of 2 x 2 x 2 mm 3 . Finally, all normalized images were spatially smoothed with a 10-mm full-width half-maximum Gaussian kernel, and single-subject statistical contrasts were computed by using the general linear model, including 3 orthogonal basis functions (canonical HRF, its time derivative and dispersion as implemented in SPM5; [START_REF] Friston | Event-related fMRI: characterizing differential responses[END_REF]. Only the estimate for the canonical HRF was used for the second level statistics, which is a measure for the amplitude of the brain response. Lowfrequency noise was removed with a high-pass filter (time constant 128 s). We modeled the 4 experimental conditions (arm idiomatic, leg idiomatic, arm literal and leg literal sentences) with the onset of the HRF response time-locked to the onset of the critical words of the sentences (i.e. early analysis window) and to a point delayed by 3 s from their offset (i.e. late analysis window). The probe sentences were modeled as separate events, though the corresponding data were not analyzed. Results are presented for both analysis windows. Group data were analyzed with a random-effects analysis. For visual display, Figures report results at p = .001, uncorrected. Tables report activations that passed the threshold of p = .001, uncorrected; activations that survived FDR correction [START_REF] Genovese | Thresholding of statistical maps in functional neuroimaging using the false discovery rate[END_REF] at p < .05 are also indicated. Stereotaxic coordinates for voxels with maximal z values within activation clusters are reported in the MNI standard space (which resembles very closely the standardized space of [START_REF] Talairach | Co-planar stereotaxic atlas of the human brain[END_REF] see Brett et al. 2002a).

Several statistical analyses aimed at testing different hypotheses were performed.

First, to assess whether reading of idioms and literal sentences activated a common cortical network and/or possibly additional selective brain areas, we carried out an analysis with 7 regions-of-interest (ROIs; "General ROI analysis"). On the basis of random-effects analysis in the early analysis window, we defined 7 ROIs activated by all sentences compared to the baseline (see Table 2; left perisylvian language areas -i.e. inferior frontal gyrus BA 45, middle temporal gyrus BA 22, angular gyrus BA 39, (known as being involved in idiom processing, see [START_REF] Lauro | Idiom comprehension: a prefrontal task?[END_REF]) and right cerebellum). This definition was done by using MARSBAR software utility (Brett et al. 2002b). For each subject and each of these 7 ROIs, average parameter estimates over voxels were calculated for spheres of radius 10 mm. This was done in both early and late analysis windows. Cortical activity in the 7 ROIs was compared between the early and late windows using a four-way ANOVA with the design Time-Window (early vs. late) x ROI x Idiomaticity (idiomatic vs. literal) x Body Part (arm vs. leg). To further analyze potential interactions, time-windows were then examined separately with a three-way ANOVA (ROI x Idiomaticity x Body Part). Significant effects are reported in the text only if they survived Greenhouse-Geisser correction.

Second, we directly tested the hypothesis that both idioms and literal sentences that include action words activate the motor cortex somatotopically. To this aim, we performed an analysis including the 2 ROIs selected from the motor localizer task ("motor localizer ROI analysis"). Because idiomatic and literal sentences elicited mainly left-lateralized activity, we selected left-hemispheric ROIs from the right finger and right foot conditions only. Right finger movements yielded activity in the left postcentral gyrus (BA 2: -50 -22 44, t(17) = 9.54), whereas activity in a left dorsal area on the midline was observed during execution of right foot movements (BA 6: -4 -26 72, t [START_REF] Smet | The cerebellum and language: the story so far[END_REF] = 16.34). Note that these postcentral activations (for which maximal t-values were obtained) may be related to somatosensory self-stimulation during motor performance (see also Hauk et al. 2004). Activity in these regions was compared between the early and late analysis windows with a four-way ANOVA (Time-Window x ROI x Idiomaticity x Body Part). Values were then subjected to a three-way ANOVA (ROI x Idiomaticity x Body Part) where the two time-windows were analyzed separately.

Since ROIs from the motor localizer ended up in postcentral cortex, we subsequently performed an additional statistical analysis with a priori selected ROIs along the central sulcus and precentral gyrus ("motor strip ROI analysis"; as in Pulvermüller et al. 2006).

A chain of 9 spheres with a radius of 10 mm was aligned along the central sulcus of the standard MNI brain between vertical z-coordinates 25 mm and 76 mm. An additional line of 9 regions just 1 cm anterior to the central ROIs was defined in the same way in the precentral gyrus. These central and precentral regions were selected a priori as belonging to the motor strip [START_REF] Penfield | The cerebral cortex in man[END_REF]; see also Pulvermüller et al. 2006 for similar methods). The subdivision of precentral and central cortex resulted in an array of 2 x 9 = 18 regions, for each of which activation values were obtained for each condition (idiomatic vs. literal, and arm-vs. leg-action relatedness) and subject. A four-way ANOVA with the design Time-Window (early vs. late) x Dorsality (9 regions, inferior to superior) x Idiomaticity x Body Part was used to compare cortical activity along the motor strip between both analysis windows. Activity was then analyzed in each of these windows with an additional four-way ANOVA (Frontality (precentral vs.

central) x Dorsality x Idiomaticity x Body Part).

Results

Behavioral results

To ensure that the 18 participants were attentive to the silent reading task, they were asked to answer yes/no questions about probe sentences by pressing one of two buttons with left index or middle finger. Mean error rate was small (8.3 %, SE = 2.45), indicating that they paid attention to the sentences.

fMRI general activation

In both early and late analysis windows, comparison of all sentences (literal and idiomatic) to the baseline (hash-mark strings) revealed left-lateralized activation in core language areas, that is, the inferior frontal gyrus (IFG) and the middle temporal gyrus (MTG; Table 2 andFigure 2). Activity was also observed in left fusiform gyrus (FG), left angular gyrus (AG), left temporal pole (TP), left dorsolateral prefrontal cortex (DLPC) and right cerebellum. Regions in the primary motor and premotor cortex were further activated during reading of action-related sentences. As can be seen from Figure 2, this activity along the motor strip tended to be more distributed and to extend further in dorsal motor areas (z coordinates > 50 mm) in the late analysis window.

-Table 2 about here --Figure 2 about here -

Idiomatic vs. literal sentence processing

The results broken down for the idiomatic and literal sentences, compared to the baseline, are presented in Figures 3a andb respectively. A common network of cortical activity was observed for both conditions in both analysis windows, with the idioms eliciting overall more distributed activation. This network included core perisylvian language areas as well as the right cerebellum (Table 3). Importantly, the precentral and middle frontal gyri including the premotor and motor cortex were activated when both literal and idiomatic sentences were being processed.

Direct comparisons between the two activation conditions showed that literal sentences failed to elicit stronger activation than idioms in any brain area. In contrast, stronger activation to idioms than to literal sentences was seen in IFG (pars triangularis of Broca's area) in both early and late analysis windows, in the pars opercularis of Broca's area in the early window only, and in MTG, right cerebellum and DLPC in the late window only (Figure 3c and Table 3).

-Table 3 about here 4.46, p = .002]) also emerged, suggesting stronger activity for idioms than for literal sentences, especially in IFG, MTG, TP and AG in the early analysis window (p-values < .025). In the late analysis window, the right cerebellum was strongly activated also showing enhancement of activity to idioms (p = .001). There was a significant ROI x Body Part interaction (early: [F (1, 17) = 3.17, p = .02]; late: [F (1, 17) = 3.56, p = .01]), documenting that activity in the 7 defined ROIs depended on the body part reference of the action verbs. In the early analysis window, arm-related sentences generally activated more strongly the defined ROIs than leg-related sentences (p-values < .035; with exceptions however in the IFG and DLPC). In the late window, activity was stronger for arm sentences than for leg sentences in the AG (p = .017).

-Figure 4 To analyze this triple interaction, activity in finger and foot areas was examined separately for early and late analysis windows (see Imaging methods). Whereas the early analysis window failed to show any ROI x Body Part interaction, the late window revealed a significant interaction of these two factors [F (1, 17) = 7.42, p = .014]. This result shows that sentences including leg-related action words elicited stronger activity in the left foot dorsal area, while arm-related sentences recruited more strongly the left finger lateral area. Note that there was overlap, but not absolute congruency, between movement and action-sentence-related focal activations in the motor system, an observation which is in very good agreement with previous work on single action words (Hauk et al. 2004a;Kemmerer et al. 2008) where also overlap, but not exact congruency, between the brain loci processing movements and action-related language was found (Hauk et al. 2004 reported the highest t-value to arm words at z = 48 mm but that for finger movements at z = 60 mm, 12 mm dorsal to it). A significant ROI x Idiomaticity interaction [F (1, 17) = 7.99, p = .012] finally documented that idioms produced greater activation than literal sentences particularly in the left foot dorsal area. Note again that the ROIs from the motor localizer were however located in the postcentral cortex, possibly due to somatosensory self-stimulation during movement execution (for discussion, see also Hauk et al. 2004a).

-Figure 4 about here -To further examine semantic somatotopy to arm-and leg-related sentences, we carried out an additional analysis for both analysis windows in regions along the motor strip ("motor strip ROI analysis", see Imaging Methods). A four-way ANOVA (Time-Window x Dorsality x Idiomaticity x Body Part) showed a significant interaction between Time-Window, Dorsality and Body Part [F (8, 136) = 4.09, p < .001], documenting that the body part reference of action words modulated cortical activity in motor areas differently for the two time-windows. Given that no significant indication of somatotopy was found in the early analysis window with the "motor localizer ROI analysis" (see above), the "motor strip ROI analysis" including the chain of 2x9 regions along the motor strip was performed for the late window only. = 23.61, p < .001], suggesting that activity along the central sulcus and in the precentral gyrus was differentially modulated by the idiomatic nature of the sentences as well as by the body part reference of the component action words. This interaction, which was not influenced by body part representations (superior vs. lateral ROIs), was due to generally enhanced BOLD signals to idioms in precentral cortex and an additional tendency for leg-idiomatic sentences to more strongly activate central areas.

-Figure 6 about here -

Discussion

Silent reading of sentences including action words activated a range of left perisylvian fronto-temporal areas with a well-known role in language processing, along with the fusiform gyrus and the right cerebellum. Idioms activated most of these areas more strongly than literal sentences, both in early and late analysis windows. Activity patterns critically depended on the body part reference of action-related words embedded into both idioms and literal sentences. Semantic somatotopy with stronger dorsal motor cortex activation for "leg action" idioms ("He kicked the habit") and relatively stronger lateral motor cortex activation for "arm action" idioms ("He grasped the idea") was evident, especially when the modeled metabolic response was adjusted to a time period after critical word ending, to capture the metabolic indexes of sentencelevel meaning processing (Figures 5 and6). These results establish for the first time the differential involvement of motor and premotor cortex in idiom processing and support theories that view abstract semantics as grounded in action-perception systems [START_REF] Barsalou | Grounded cognition[END_REF][START_REF] Glenberg | Language and action: creating sensible combination of ideas[END_REF][START_REF] Martin | The representation of object concepts in the brain[END_REF][START_REF] Pulvermüller | Brain mechanisms linking language and action[END_REF].

Materials putting a particularly heavy burden on the language system are known to activate the left fronto-temporal language network more strongly and in a more widespread fashion than relatively simple language stimuli. For instance, it is well known that the N400 brain response is enlarged to sentences including semantically unexpected constituent words [START_REF] Kutas | Brain potentials during reading reflect word expectancy and semantic association[END_REF][START_REF] Van Berkum | Semantic integration in sentences and discourse: evidence from the N400[END_REF], the main generators of this effect being localized in the posterior perisylvian cortex [START_REF] Van Petten | Neural localization of semantic context effects in electromagnetic and hemodynamic studies[END_REF]. [START_REF] Rodd | The neural mechanisms of speech comprehension: fMRI studies of semantic ambiguity[END_REF] presented sentences with unexpected ambiguities and found enhanced and more distributed fronto-temporal fMRI activation relative to unambiguous control sentences. [START_REF] Lauro | Idiom comprehension: a prefrontal task?[END_REF] probed literal and idiomatic sentences and found left fronto-temporal activation also extending into anterior inferior frontal cortex, anterior temporal cortex and angular gyrus (see also [START_REF] Lee | Metaphorical vs. literal word meanings: fMRI evidence against a selective role of the right hemisphere[END_REF][START_REF] Rapp | Neural correlates of metaphor processing[END_REF][START_REF] Zempleni | Evidence for bilateral involvement in idiom comprehension: An fMRI study[END_REF] for fMRI studies on idiom and metaphor processing). Our present activity enhancements to idioms at left inferior frontal and middle temporal sites are consistent with this pattern of results. They may, however, be best explained as an index of increased workload on the language system rather than as specific brain signature of idiom processing.

In the late analysis window examined here, we found additional idiom-related activation enhancement in the cerebellum and in the middle frontal gyrus extending into frontocentral motor and premotor cortex. The stronger cerebellar activity to idioms extends previous findings on the role of this structure in perceptual and language processing [START_REF] Ackermann | Cerebellar contributions to speech production and speech perception: psycholinguistic and neurobiological perspectives[END_REF]Braitenberg et al. 1997;[START_REF] Smet | The cerebellum and language: the story so far[END_REF][START_REF] Ivry | Timing functions of the cerebellum[END_REF]. The co-occurrence of cerebellar and motor cortex increased activation to idiomatic sentences further suggests a consorted role of these structures in motor cognition [START_REF] Jeannerod | Motor cognition: What action tells the self[END_REF]) brought about by abstract action-related language.

For both idiomatic and literal sentences, we observed differences that reflected the meaning of their constituent arm-and leg-related action words. This influence became evident in a range of analyses, most notably with the significant ROI x Body Part interaction in the analysis of motor and premotor cortex activation ("motor strip ROI analysis"; Figure 6). This interaction did not involve the "Idiomaticity" factor, thus documenting that the well-known semantic somatotopy found for concrete action words and sentences [START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF]Hauk et al. 2004;Pulvermuller et al. 2001;Tettamanti et al. 2005) can be replicated for abstract sentences including action words.

Activation of frontocentral motor and premotor areas was relatively weak at the onset of critical words but was strong after their offset (Figure 5). If the fMRI brain response reflecting sentence meaning is delayed relative to that of single words [START_REF] Humphries | Time-course of semantic processes during sentence comprehension: An fMRI study[END_REF], this late activation of motor areas can be linked to the sentence processing stage.

Any contribution of individual action words would have been expected to arise at action word onset (on average 1.2 s before critical word onset and thus 4.7 s before the late analysis window) and to decrease with time. Our results therefore suggest that the orchestration of abstract meaning in the human brain is not solely explained by the activation of unspecific semantic centers in fronto-temporal cortex, but that it involves late complementary activations in the sensory-motor system. These referentially grounded activations may play a specific functional role in the composition of sentence meaning. However, further work using techniques such as transcranial magnetic stimulation (TMS) and neuropsychological studies in brain-damaged patients are necessary to draw firm conclusions on functional contributions of the motor system to idiom comprehension.

The present results support a compositional perspective on semantic processing postulating that idiom meaning is computed from the semantics of constituent words and from combinatorial information. Semantic somatotopy to idioms indeed suggests that meaning aspects of words included in these sentences are being re-accessed and combined in the relatively late construction of sentence meaning. Access to concrete referential aspects of constituent words, as it regularly occurs in language comprehension, appears not as an irrelevant by-product but rather as an important step in the comprehension process, which may play a role in the comprehension of figurative language too [START_REF] Gibbs | Speakers' assumptions about the lexical flexibility of idioms[END_REF][START_REF] Gibbs | Idioms and mental imagery: The metaphorical motivation for idiomatic meaning[END_REF][START_REF] Titone | On the compositional and noncompositional nature of idiomatic expressions[END_REF]. In the context of the present study, one may argue that the activation of dorsal and lateral motor and premotor cortex was related to the processing of leg-and arm-related words per se and not to the comprehension of sentence meaning. We should remind the reader, however, that somatotopic semantic grounding of constituent arm/leg words in lateral/dorsal frontocentral cortex, respectively, was relatively weak at action word onset and also at presentation of critical words (Figure 5). It became pronounced about 3 s after sentence ending, suggesting its specificity to a late stage of sentence processing. At such a late stage, it would be extremely unlikely that one particular word from the several ones included in the sentence is still processed in depth in isolation and dominates the brain response. Rather, it appears plausible that semantic integration at the sentence level underlies metabolic changes, which might occur especially late for highly abstract sentences. This pattern of results is therefore consistent with a gradual emergence of semantic somatotopy in the processing of idiom meaning and is not explained by word-related activation.

Previous electrophysiological studies have shown instant spreading of activity to motor regions during action word recognition (< 200 ms; Boulenger et al. 2008b;Hauk and Pulvermüller 2004;[START_REF] Kiefer | Experience-dependent plasticity of conceptual representations in human sensory-motor areas[END_REF][START_REF] Pulvermüller | Nouns and verbs in the intact brain: evidence from event-related potentials and high-frequency cortical responses[END_REF]Pulvermüller et al. , 2005b; for a review, see [START_REF] Hauk | The time course of action and action-word comprehension in the human brain as revealed by neurophysiology[END_REF]. Despite the apparent discrepancy between these results and those of the present study, the reader should however be reminded that 1/ previous neurophysiological work focused on single words while we here used complex sentences, where action words had to be integrated into their context; 2/ as opposed to fMRI, electrophysiological techniques offer high temporal resolution that allows precise tracking of the time-course of brain activation during cognitive processes. Future research, using EEG and/or MEG for instance, is therefore necessary to address the question of when grasping ideas activates the motor system. Comparing the present results with previous fMRI studies of brain activation to action-related sentences [START_REF] Aziz-Zadeh | Congruent embodied representations for visually presented actions and linguistic phrases describing actions[END_REF]Tettamanti et al. 2005), it is still noteworthy that BOLD signal changes occurred relatively late in the present study. This indicates that, if idioms appear in the context of literal sentences, as they do in normal language use, semantic and any post-understanding processes may be delayed relative to an experimental context where only literal sentences are presented. We note however that early brain reflections of semantic processing of word pairs [START_REF] Hoenig | Conceptual flexibility in the human brain: dynamic recruitment of semantic maps from visual, motor, and motion-related areas[END_REF][START_REF] Shtyrov | Early activation dynamics in the left temporal and inferior-frontal cortex reflect semantic context integration[END_REF] have recently been reported early-on (100-150 ms) and even the first effects of sentence level semantics have been found between 100-200 ms [START_REF] Penolazzi | Early semantic context integration and lexical access as revealed by event-related brain potentials[END_REF][START_REF] Sereno | Context effects in word recognition: evidence for early interactive processing[END_REF]. It may therefore be suggested that the idiomaticity of sentence stimuli in the present experiment critically contributed to the lateness of the haemodynamic brain response. To further explore the temporal structure of idiom comprehension, we would like to re-emphasize the need for future neurophysiological work.

The late effects we observed here also raise the issue that motor activity could be epiphenomenal with respect to sentence comprehension or could reflect motor imagery after semantic access. It is indeed possible that post-understanding processes [START_REF] Glenberg | Grounding language in action[END_REF], following semantic sentence level analysis, are reflected (e.g. imagining a picture or a scene matching the sentence content). If this is true, semantic somatotopy of this secondary process triggered by sentence meaning would still argue against an abstract symbolic perspective. Indeed, in such an abstract symbolic framework, the meaning of the phrase "grasp an idea" is semantically unrelated to grasping. The only way to account for such secondary somatotopic activation would be through semantic somatotopy in sentence meaning analysis and consequent somatotopy of the secondary (imagery or the like) process.

Models assuming storage of idiom meanings as whole units unrelated to the meaning of their constituent words do not provide an explanation of the observed differences between idioms that include arm and leg action words. To provide such an explanation, two assumptions are necessary: 1/ Idiom meaning must be computed on-line from the meaning of constituent words, and 2/ Semantic aspects grounded in action-perception knowledge must play a critical role in the composition process yielding idiom meaning [START_REF] Gibbs | Idioms and mental imagery: The metaphorical motivation for idiomatic meaning[END_REF]. In this sense, the present results support both semantic compositionality and the grounding of figurative/abstract language in concrete sensorymotor information and in their corresponding specific brain circuits. Motor systems of the brain, including motor and premotor cortex, and the motor cognitions they process [START_REF] Jeannerod | Motor cognition: What action tells the self[END_REF]) appear to be central for understanding idioms. When "grasping ideas", the motor system is engaged in a specific manner.

--

  Figure 3 about here -General ROI analysis To test activity dynamics related to Time-Window, Idiomaticity and Body Part reference of action verbs, along with their possible interactive effects, activity in 7 ROIs was compared between arm/leg-related idiomatic/literal sentences in the early and late analysis windows ("General ROI analysis", see Imaging Methods). A four-way ANOVA (Time-Window x ROI x Idiomaticity x Body-Part) revealed a significant main effect of Time-Window [F (1, 17) = 18.09, p = .001] and a significant Time-Window x ROI interaction [F (1, 17) = 12.31, p < .001], indicating that cortical activity was weaker in the late than in the early analysis window in a range of areas (IFG, MTG, FG, TP and AG, p-values < .003), whereas in other regions (cerebellum and DLPC), no significant change was observed (i.e. prolonged activation). To further analyze these complex interactions, the analysis windows were examined with separate three-way ANOVAs (see Imaging Methods). Results, which are presented in Figure 4, first revealed a significant main effect of ROI (early window: [F (1, 17) = 17.24, p = .001]; late window: [F (1, 17) = 11.65, p = .001]), indicating that cortical activity was particularly strong in left perisylvian areas (IFG, MTG and AG). A significant main effect of Idiomaticity (early: [F (1, 17) = 11.92; p = .003]; late: [F (1, 17) = 16.42, p = .001]) and a significant ROI x Idiomaticity interaction (in the early window only: [F (1, 17) =

  about here -Semantic somatotopy Inspection of Figure5shows that in the critical comparison between idioms including arm-and leg-related action words, different activation patterns were obtained. Already in the early analysis window, only idioms that include leg words led to spreading of activity to leg areas in dorsocentral cortex compared to the baseline (RH: 10 -24 60, t[START_REF] Smet | The cerebellum and language: the story so far[END_REF] = 7.1; LH: -22 -26 60, t (17) = 5.3). This effect seemed even more pronounced in the late window (LH: -18 -28 72, t (17) = 7.19; RH: 22 -26 70, t (17) = 9; LH: -12 -20 72, t (17) = 7.71; RH: 20 -12 68, t (17) = 8.59). A four-way ANOVA (Time-Window x ROI x Idiomaticity x Body Part) including the 2 ROIs selected from the motor localizer experiment ("motor localizer ROI analysis", see Imaging Methods) revealed a significant interaction between Time-Window, ROI and Body Part [F (1, 17) = 7.6, p = .013]. This establishes that cortical activity in motor areas was modulated by the body partrelationship of words included in the sentences differently for the two time-windows.

  A four-way ANOVA (Frontality x Dorsality x Idiomaticity x Body Part) revealed a significant interaction between the Dorsality and Body Part factors, [F (8, 136) = 4.40, p < .001]. As can be seen in Figure 6, in dorsal ROIs (z-coordinate ~ 75 mm), leg sentences elicited stronger activation than arm sentences, and the opposite pattern, relatively stronger arm sentence activation, was seen in lateral ROIs (25 mm ≤ z ≤ 50 mm). Semantic somatotopy (Dorsality x Body Part interaction) was also found in separate analyses for central (p = .002) and precentral regions (p = .05), suggesting that both motor and premotor cortex contributed to this effect. A two-way ANOVA (ROI x Body Part), including a dorsal motor ROI (computed over the 3 regions with 73 ≤ z ≤ 76 mm) and a lateral motor ROI (computed over the 3 regions with 35 ≤ z ≤ 52 mm), confirmed this result by showing a significant ROI x Body Part interaction [F (1, 17) = 6.55, p = .02]. Finally, additional analyses on local differences revealed stronger leg-than arm-sentence responses in the most dorsal ROI (z = 75 mm; p = .033), whereas stronger activity for arm sentences was found in a more lateral ROI (z = 44 mm; p = .037). These regions are in good agreement with those reported by Hauk et al. (2004a) who found maximal activation probabilities (t values) in precentral gyrus for arm words at z = 48 mm and for leg words at z = 64 mm.A significant Frontality x Idiomaticity x Body Part interaction also emerged[F (8, 136)

Figure 1 :

 1 Figure1: Design of the experiment. Each trial was composed of 10 displays/screens, here represented by grey boxes, where the consecutive stimuli -fixation cross "+" and words making sentences -appeared each for 500 ms. Two examples of arm-and legaction-related sentences are given. SOA between two consecutive critical words (indicated in bold) was 6.6 s. A fixed delay of 2.1 s, where a fixation cross remained on the screen, was inserted between two consecutive trials, so that the inter-sentence interval (ISI) varied between 2.6 and 5.1 s. The oblique axis on the right illustrates the temporal sequence of the trials and gives the onset of the corresponding stimulus (in milliseconds). TR of the EPI sequence is also represented (TR = 2 s).

Figure 2 :

 2 Figure 2: Cortical activation during silent reading of all sentences (idiomatic and literal) relative to the baseline (hash-mark strings) in the (a) early and (b) late analysis windows (p < .001, uncorrected). Results are rendered on a standard brain surface. Top panel: lateral view of the brain, Bottom panel: top view. Note the greater activation of precentral areas in the late window.

Figure 3 :

 3 Figure 3: Cortical activation during silent reading of (a) idioms and (b) literal sentences (p < .001, uncorrected), compared to the baseline (hash-marks strings), in the early (top panel) and late analysis windows (bottom panel). Results are rendered on a standard brain surface. Specific activations for the direct contrast between idioms and literal sentences are reported for both windows in (c). The inset in (c) highlights the specific activation observed in the right cerebellum for idioms, compared to literal sentences, in the late window (bottom panel).

Figure 4 :

 4 Figure 4: Mean parameter estimates (in arbitrary units) for the 7 ROIs in the 4 experimental conditions (arm idiomatic, leg idiomatic, arm literal and leg literal sentences) in the (a) early and (b) late analysis windows. Error bars are reported. LDLPC, left dorsolateral prefrontal cortex; LIFG, left inferior frontal gyrus; LTP, left temporal pole; LMTG, left middle temporal gyrus; LAG, left angular gyrus; LFG, left fusiform gyrus; RCrbllm, right cerebellum.

Figure 5 :

 5 Figure 5: Somatotopic activation for idioms including arm-(in red) and leg-related action words (in blue), compared to the baseline (hash-mark strings), in the early (left panel) and late analysis windows (right panel; p < .001, uncorrected). Results are rendered on a standard brain surface.

Figure 6 :

 6 Figure 6: Semantic somatotopy for literal and idiomatic sentences along the motor strip in the late analysis window. Bar graphs show mean parameter estimates (in arbitrary units) for the 9 ROIs aligned along the central sulcus and the precentral gyrus that are reported for sentences including arm-(in red) and leg-related action words (in blue). For each graph, the /x/ and /z/ coordinates are indicated at the top right (x, z). The locations of the ROIs are reported (yellow circles) on a coronal slice of the brain. Somatotopic activations elicited during finger (in red) and foot movements (in blue) during the localizer experiment are also shown.

  Figure 1

Table 1 :

 1 Mean values of word frequency, lemma frequency, length in letters, number of syllables, bigram frequency, trigram frequency and number of orthographic neighbours are reported for idiomatic and literal critical words of the sentences, and for arm-and leg-related action verbs. P-values for ANOVAs (by items) are reported.

Table 2 :

 2 Coordinates and statistics for activation peaks produced by all sentences (idiomatic and literal) vs. the baseline (hash-marks strings) in the (a) early and (b) late analysis windows. The corresponding Brodmann areas (BA) are reported in the second column of the table. The 7 ROIs selected on the basis of random-effects analysis in the early analysis window, which were included in the "General ROI analysis", are highlighted in grey in Table1a. All clusters passed the significance threshold at p < .001, uncorrected, and survived FDR (False Discovery Rate) correction at p < .05. LH, left hemisphere; RH, right hemisphere.

Table 3 :

 3 Coordinates and statistics for activation peaks produced by literal and idiomatic sentences in the (a) early and (b) late analysis windows, compared to the baseline (hashmarks strings). Activation peaks for the contrast idiomatic vs. literal stimuli are also reported. The corresponding Brodmann areas (BA) are indicated in the second column of the table. All clusters passed the significance threshold at p < .001, uncorrected.Activations that did not survive FDR correction at p < .05 are indexed by asterisks. LH, MNI coordinates along with T-values are given for the maximally activated voxel in each local cluster. All clusters passed the significance threshold at p < .001, uncorrected, and survived FDR correction at p < .05.

	left hemisphere; RH, right hemisphere.

TABLE 3 ( a )

 3a MNI coordinates along with T-values are given for the maximally activated voxel in each local cluster. All clusters passed the significance threshold at p < .001, uncorrected. Activations that did not survive FDR correction at p < .05 are indexed by asterisks. MNI coordinates along with T-values are given for the maximally activated voxel in each local cluster. All clusters passed the significance threshold at p < .001, uncorrected. Activations that did not survive FDR correction at p < .05 are indexed by asterisks.

	(b)
	LATE ANALYSIS WINDOW
	Brain Region	MNI x y z		T (17)
	Brain Region Literal sentences Inferior frontal gyrus EARLY ANALYSIS WINDOW BA 45 LH -48 18 12 MNI x y z BA 47 LH -50 20 0 Literal sentences Inferior frontal gyrus BA 45 LH -50 20 16 Middle frontal gyrus BA 47 LH -48 36 -6 BA 47 LH -48 30 2 BA 6 LH -40 4 48 Middle frontal gyrus BA 6 LH -40 -2 48 BA 46 LH -50 24 26 BA 9 LH -38 10 24 Middle temporal gyrus BA 22 LH -52 -40 2 Superior frontal gyrus BA 6 LH -4 10 54 BA 21 LH -60 -48 6 BA 6 LH -6 12 70 Superior temporal gyrus BA 22 LH -58 -54 12 Middle temporal gyrus BA 21 LH -54 -6 -14 BA 21 LH -50 -22 -6 BA 39 LH -56 -58 12 Cerebellum RH 16 -80 -34	T (17) 5.74 6.29 4.98 7.49 7.63 6.37 5.85 8.71 6.91 4.81 5.98 7.27 5.48 4.29 5.12 7.08 6.82 6.44
	Superior temporal gyrus	BA 22 LH -64 -46 8	7.2
		BA 38 LH -52 12 -18	5.61
	Fusiform gyrus Idiomatic sentences Cerebellum Inferior frontal gyrus Middle frontal gyrus	BA 37 LH -42 -40 -16 RH 12 -78 -34 BA 45 LH -48 22 12 BA 47 LH -44 36 -4	7.96 10.49 8.46 10.68
	Idiomatic sentences Inferior frontal gyrus	BA 46 LH -52 24 4 BA 45 LH -48 30 4 BA 6 LH -34 -4 54	11.65 9.52 7.34
	Medial frontal gyrus	BA 47 LH -36 32 -16 BA 8 LH -6 42 40	6.68 5.18
	Superior frontal gyrus	BA 46 LH -52 26 12 BA 6 LH -12 18 60	8.74 6.93
		BA 9 LH -36 10 24 BA 9 LH -12 52 28	5.97 4.93
	Medial frontal gyrus Middle temporal gyrus	BA 6 LH -6 -14 70 BA 21 LH -52 -28 -4	3.89 6.78
	Superior frontal gyrus	BA 6 LH -4 12 58 BA 22 LH -56 -40 2	7.38 8.87
		BA 9 LH -8 50 30 BA 39 LH -58 -58 20	5.9 11.01
	Middle temporal gyrus Fusiform gyrus	BA 21 LH -54 -6 -16 BA 37 LH -46 -36 -10	8.37 6.79
	Precentral gyrus	BA 39 LH -56 -60 12 BA 6 RH 16 -18 68	7.02 8.17
	Superior temporal gyrus Cerebellum	BA 22 LH -64 -46 8 RH 16 -82 -30	7.78 10.19
		BA 38 LH -50 12 -22	7.5
	Precentral gyrus Idiomatic > Literal	BA 6 LH -42 -4 48	7.59
	Inferior frontal gyrus	BA 6 LH -38 0 BA 45 LH -44 30 2 * 4.84 32 7.42
	Cerebellum Middle frontal gyrus	RH 12 -78 -34 BA 9 LH -56 20 26 * 4.37 7.72
	Middle temporal gyrus	BA 21 LH -62 -56 6 * 4.69
	Idiomatic > Literal Cerebellum	LH 20 -82 -32 * 5.13
	Inferior frontal gyrus	BA 45 LH -48 28	6 * 5.61
		BA 44 LH -58 16 10 * 4.78

Acknowledgements

We thank Elisabeth Fonteneau and Kambiz Tavabi for helpful comments on a previous version of this paper, as well as Matthew H. Davis for his help with the SPM analysis.

We also thank four anonymous Referees for their efforts and help in improving this work. This work was supported by the Medical Research Council (UK) (U1055.04.003.00001.01, U1055.04.003.00003.01) and by the European Community under the "Information Society Technologies Programme" (IST-2001-35282) and the "New and Emerging Science and Technologies" Programme (NEST, Nestcom Project) to F.P. V.B. was supported by a postdoctoral fellowship from the Fyssen Foundation.


Tomohiro Ito

Atsuhiko Shintani

Chihiro Nakagawa

VIBRATION CHARACTERISTICS OF A PERFORATED PLATE IMMERSED IN FLUID

Keywords: Hole diameter f d, Damped natural frequency f n, Natural frequency of plate f w, Natural frequency in water h, Thickness of plate k, Modal stiffness m, Mass of square plate Natural angular frequency

Perforated plates are used in many mechanical structures in thermal power plants, nuclear power plants, or chemical plants etc. Cylindrical structures made by the perforated plates are also found in many places. However, vibration characteristics of the structures made by perforated plates are not fully clarified, especially for the structures immersed in liquid. The stiffness of the structures becomes smaller than that of ones made by simple plates with no holes, while the mass of the structures also becomes smaller. According to the balance between the stiffness reduction and mass reduction, natural frequencies will be decided. Moreover, added mass and added damping effects are very large in liquid, and are thought to largely change due to holes.

In this study, as a fundamental step, a perforated plate is treated. The vibration characteristics such as natural frequency and damping ratio are studied for various hole numbers or various opening ratios by both numerical simulations and simple test models. Vibration tests are conducted in liquid as well as in air.

INTRODUCTION

In various industrial plants such as chemical plants, thermal power plants, nuclear power plants and food industries, perforated plates are often used for reducing weight or for effective mixture. Another example is exhaust pipes of automobile which is used for reducing noise. And cylindrical structures made by the perforated plates are also found in many places such as marine structures. However, vibration characteristics of the structures made by perforated plates are not fully clarified, especially for the structures immersed in liquid. The stiffness of the structures becomes smaller than that of ones made by simple plates with no holes, while the mass of the structures also becomes smaller. According to the balance between the stiffness reduction and mass reduction, natural frequencies will be decided. Moreover, added mass and added damping effects are very large in liquid, and are thought to largely change due to holes. Thus, vibration characteristics and vibration response of the perforated plates will change largely in liquid. Therefore, it is very important to clarify the stiffness reduction, mass reduction, added mass and added damping of a perforated plate immersed in liquid.

As for the natural frequencies of a square plate with a hole in air, Kumai, T. proposed a modification ratio between natural frequencies of a plate without and with a hole [START_REF] Kumai | Flexural vibration of square plate with a central circular hole[END_REF]. Huang, M. et al. studied natural frequencies of a square plate in air with a various-shaped hole by approximated calculation [START_REF] Huang | Free vibration analysis of rectangular plates with variously-shaped holes[END_REF]. Cheng, L. discussed natural frequencies of a circular plate in air with a hole with eccentricity [START_REF] Cheng | Vibration analysis of annular-like plates[END_REF]. As for natural frequencies of a perforated plate in air, Okitsu, A. et al. studied the effects of hole pitch theoretically and experimentally [START_REF] Okitsu | Experimental study on the dynamic characteristics of perforated plates[END_REF]. And Matsumoto, A. et al. studied added mass of liquid of a square plate without holes theoretically and experimentally [START_REF] Matsumoto | Consideration on added mass acting on rectangular plate[END_REF]. However, these studies do not treat the vibration characteristics of a perforated plate immersed in liquid. Therefore, it is thought that systematic investigation will be necessary for using perforated plates in the actual design.

In this paper, added mass, added damping of liquid acting on a perforated plate and vibration characteristics of a perforated plate are investigated for various opening ratios and hole numbers analytically and experimentally. A square plate is treated here. In the analyses, natural frequencies are calculated for the perforated plate in air using a general-purpose computer software ANSYS. In the experiments, natural frequencies and damping ratios are measured both in air and in water. From these data, modal stiffness, added mass and added damping are evaluated.

ANALYSIS Analytical Model

In this paper, a square plate clamped at one end is treated as shown in Fig. 1. The length of the plate is defined as a, and the diameter of a hole is defined as d.

Figure 1 Square plate model with a hole

Here, natural frequencies in air are calculated for various opening ratio R (=A/A') and hole numbers N. A is the area of a square plate without holes, and A' is the area of a hole. Opening ratio is varied from 0 to 50 %. And for R=20%, the hole number is varied 1 to 36. The relationship between the opening ratio and the ratio of d/a is shown in Table 1. And the finite element mesh figures for each case are shown in Figs. 2 and3. The dimensions of a square plate are selected as; a is 200 mm, thickness h is 1mm.

Table 1 Opening ratio of a plate

In this paper, an aluminum plate is treated, thus,  is 2690 kg/m 3 , E is 73 Gpa and  is 0.3.

Analytical Results

Vibration mode and natural frequency for various opening ratios

Vibration mode and natural frequency in air for various opening ratio are calculated. Figure 4 shows the mode shapes from 1 st to 5 th for the opening ratio of R =0, 13, 50 %. Major differences among 3 opening ratios are not found for the mode shapes from 1 st to 3 rd . But slight differences are found for the higher mode shapes such as 4 th and 5 th among 3 opening ratio.

Figure 5 shows the natural frequencies for various opening ratios. Natural frequencies except for 4 th mode shows the tendency to decrease according to the increase of opening ratio. The increase in the opening ratio means the reduction in the stiffness and mass.

From these results, it can be said that the stiffness reduction is slightly larger than that of mass reduction. Only 4 th mode shows a different tendency, where natural frequencies decrease according to the increase of the opening ratio from 0 to 13 % as well as other mode numbers, but those tend to increase for larger opening ratio than 13 %. In Fig. 4, large mode shape change is not found for 1 st to 3 rd and 5 th modes for various opening ratio, but mode shape change for 4 th mode is relatively large than that for other mode numbers. This may be the reason for different tendency of 4 th mode.

Vibration mode and natural frequency for various hole numbers

As well as the effects of the opening ratio, effects of hole numbers are studied. Figure 6 shows vibration modes from 1 st to 5 th mode for N=1, 9, 36, and natural frequencies for various hole numbers are illustrated in Fig. 7.

Major change in mode shapes are not found for N=1~36. And as shown in Fig. 7, natural frequency once reduces when the hole number increases from N=0 to N=1, but for N>1, natural frequency is almost constant.

EXPERIMENT Experimental Apparatus

At first, free vibration tests in air are conducted and after the tests in air, free vibration tests in water are conducted. In these tests, natural frequencies and damping ratios are measured, and based on these results, added mass and added damping of liquid are evaluated.

A square plate with the same dimensions as those of the analytical model is used here. In order to study the effects of opening ratio, 6 kinds of opening ratios are selected as shown in Table 2. In this table, mass of a plate is also shown. Mass reduction for the plate with opening ratio of 38 % is around 35 %. In order to study the effects of hole numbers, 6 kinds of hole numbers from 1 to 25 are selected. Here, the opening ratio is kept as 20%. Figure 8 shows photos for the test specimen. The square plate is clamped at one end to a rigid box-type channel. This channel is fixed to the base floor, and installed in a rectangular vessel as shown in Fig. 9.

Measuring Method

In order to measure frequency, water proof type strain

Experimental Results

Natural frequency in air

Time histories for damped free vibration in air are obtained. From these data, 1 st mode natural frequency is evaluated. Table 3 and4 show natural frequencies for various opening ratios and hole numbers. In these tables, modal mass and modal stiffness are also shown. These values are generally available from eigenvalue analysis by FEM code such as ANSYS. However, for a perforated plate with many holes, conducting FEM analysis requires a lot of efforts, therefore, in this study, modal mass and modal stiffness are obtained using a beam theory for convenience assuming that the mode shape in the experiment of a square plate in air is almost the same as that of a cantilever beam in air, although it is well understood that this manner is an approximation and there is surely slight difference between them. Modal mass of 1 st mode calculated based on a beam theory is found to be 0.39 times of the total mass of a square plate. And this reduction rate is assumed to be the same as those of various opening ratios and hole numbers. From modal mass and measured natural frequency, modal stiffness is also calculated. Fig. 10 and 11 illustrate the trend of natural frequency against opening ratio or hole numbers. In these figures, analytical values obtained by ANSYS are also shown by a broken line.

As for the effects of opening ratio, natural frequencies by experiments also tend to decrease according to the increase in opening ratio as well as the analytical values. As shown in Table 3, both natural frequency and modal mass decrease according to the increase of opening ratio. Modal stiffness also decreases. And as for the effects of hole numbers, the trend in the experimental values is in good agreement with that of analytical values, i.e., large reduction is observed from N=0 to N=1, but natural frequencies are almost the same for large hole numbers. Thus modal stiffness change is thought to be very small. Slight differences between the experimental and analytical values are found. The experimental values are smaller than the analytical ones. Analyses are conducted based on the perfect-clamp condition, however the actual clamped end has finite stiffness. The differences in the clamp-condition may be the main reason for the differences between the experimental and analytical values.

Damping ratio in air

Damping ratios are evaluated from the time histories of damped free vibration. Fig. 12 and 13 illustrate the relationship between damping ratio and opening ratio or hole numbers. It is known that the damping ratio in air tends to decrease slightly according to the increase in opening ratio, and are almost same for various hole numbers. But the damping ratios in air are very small, i.e., less than 0.5%. Thus, it is said that the differences in damping ratios due to opening ratio and hole numbers are very small in air.

Natural frequency in water

Time histories for damped free vibration in water are obtained. Figure 14 illustrates typical time histories for various opening ratios. From these data, natural frequency is evaluated. Tables 6 and7 show natural frequencies for various opening ratios or hole numbers. Figs. 15 and 16 illustrate the trend of natural frequency against opening ratio or hole numbers respectively. In Figs. 15 and 16, the trend in air is also plotted for comparison.

It is found that the natural frequencies decrease to 1/5~1/3 of those in air and tend to increase according to the increase of opening ratio or hole numbers. This frequency reduction indicates the existence of large added mass, because there is no added stiffness in this experiment. And it is thought that added mass tends to become smaller according to the increase in opening ratio and hole numbers. Therefore, added mass is evaluated through these data. In air

In water

Damping ratio in water

Damping ratios in water for various opening ratios and hole numbers are illustrated in Figs. 17 and 18, respectively. As is known from time histories shown in Fig. 14, damping ratio for first few cycles is larger than that for subsequent cycles. Thus, it is said that damping ratio has amplitude dependency. Therefore, damping ratio is evaluated both for large amplitude region and small amplitude region. The former is denoted as 1  and the latter is as 2  . Damping ratios for small amplitude 2  are very small and almost flat for all opening ratios. But damping ratios for large amplitude 1

 are large and tend to increase according to the increase in opening ratio or hole numbers. When we increase the opening ratio, damping ratio becomes more than 2 times larger than that for the plate without holes. And as for hole numbers, damping ratio becomes more than 3 times larger than that for the plate without holes.

Added mass

As is stated before, there is no added stiffness in this tests, then, modal stiffness is thought to be the same as those in air. Therefore, natural frequencies will be expressed by the following equation typically seen for a single degree of freedom system.

1 2 w a k f mm    (2)
Here, , dw ff are damped natural frequency and un-damped natural frequency in water respectively, and , , , a k m m  are measured damping ratio in water, modal stiffness, modal mass of a square plate and added mass of liquid acting on the plate, respectively. From these equations, added mass a m is evaluated by the following equation; (1 ) ( 2)

a d k mm f     (3)
From Tables 3,4, 6, 7 and Figs. 17 and 18, added mass of liquid acting on the square plate is evaluated using Eq. ( 3). In these calculations, 1  is employed. The relationship between added mass and opening ratio and that between added mass and hole numbers are shown in Figs. 19 and20. Tables 8 and9 show both damping ratio and added mass for large amplitude region for various opening ratios or hole numbers.

Added mass decreases according to the increase of opening ratio, and that for d/a=0.7 decreases to 1/4 of that for no hole. As for the effects of hole numbers, added mass for a single hole suddenly decreases to 1/2 of that of no hole case. And for larger hole numbers than N=1, added mass gradually decreases and becomes around 1/3 of that of no hole case.

DISCUSSIONS

In this paper, modal mass and modal stiffness in air are evaluated, and also damping ratio, damped natural frequency and added mass of liquid are evaluated in water for various In air, as for the effects of opening ratio, modal mass and modal stiffness decrease according to the increase of opening ratio, and reduction rate for modal stiffness is larger than that of modal mass. Thus, natural frequency tends to decrease according to the increase of opening ratio. As for the effects of hole numbers, natural frequency and modal stiffness also tend to decrease according to the increase of hole numbers. The reduction rate from no hole case to N=1 case is very large. But for N>1, natural frequency and modal stiffness are almost constant.

In water, on the other hand, damped natural frequency for d/a=0.7 becomes 30 % larger than that for the plate without holes, and that for N=25 becomes 54 % larger than that for N=0. This is mainly caused by the added mass reduction due to holes. When we increase opening ratio, added mass becomes 1/4 of that for the case without holes. As well as the effects of opening ratio, when we increase the hole numbers, added mass becomes 1/4 of that for the case without holes.

Another important fact is that damping ratio in water drastically increases due to the presence of holes. When we increase the opening ratio from 0 to d/a=0.7, damping ratio becomes 2.3 times larger. And also when we increase the hole numbers from 0 to 25, damping ratio becomes 3.3 times larger. Presence of a hole means the increase of edge line. Vortices will be induced around the edge and vibration energy will be absorbed by the vortices, thus damping ratio is thought to increase for the square plate with large opening ratio or large hole numbers.

The reduction in added mass means the increase in natural frequency. This fact is very important for the design for the mechanical structures made by perforated plates immersed in liquid. And damping ratio increase will very useful to maintain the structural integrity of these mechanical structures.

CONCLUSIONS

In this study, vibration characteristics such as natural frequency and damping ratio of the perforated plate were investigated.

A cantilever type square plate was treated. Natural frequency in air was analyzed by ANSYS, and modal mass and modal stiffness were evaluated. The effects of opening ratio and hole numbers were studied. And a plate model with the same dimensions as the analytical model was fabricated. Natural frequency and damping ratio were measured both in air and in water from free vibration tests. From these data, added mass of water and damping ratio were evaluated for various opening ratios and hole numbers.

The following conclusions are obtained: (1) In air, modal mass and modal stiffness decrease according to the increase of opening ratio. Modal stiffness reduction rate is larger than that of modal mass. Thus, natural frequency tends to decrease according to the increase in opening ratio. As for the effects of hole numbers, natural frequency and modal stiffness also tend to decrease according to the increase in hole numbers. The reduction rate from no hole case to N=1 case is very large. But for N>1, the effects of hole numbers are small. (2) In water, damped natural frequency for d/a=0.7 becomes 30 % larger than that for the plate without holes, and that for N=25 becomes 54 % larger than that for N=0. This indicates that the added mass reduces due to holes. Added mass for d/a=0.7 reduces to 1/4 of that for the plate without holes. Added mass for N=25 also reduces to 1/4 of that for the plate without holes. (3) Damping ratio in water drastically increases due to the presence of holes. Damping ratio for d/a=0.7 becomes 2.3 times larger than that for the plate without holes, and that for N=25 also becomes 3.3 times larger than that for the plate without holes. (4) The reduction in added mass is very important in designing the mechanical structures made by perforated plates immersed in liquid. And the increase in damping ratio will be very useful to maintain the structural integrity of these mechanical structures.

Figure 2 Figure 3

 23 Figure 2 Finite element mesh figures with a hole Fig. 4 Mode shapes for various opening ratio R

Fig. 5 Fig. 6

 56 Fig. 5 Natural frequencies for various opening ratio R

Fig. 9

 9 Fig. 9 Outline of experimental set up

Fig. 11

 11 Fig. 11 Natural frequency in air for various hole numbers

Fig. 12

 12 Fig.12 Damping ratios in air for various opening ratios

Fig. 13

 13 Fig. 15 Natural frequencies in water for various opening ratio

Fig. 16

 16 Fig. 16 Natural frequencies in water for various hole numbers

2 Fig

 2 Fig. 17 Damping ratios in water for various opening ratio

Fig. 21

 21 Fig. 21 Ratios of various parameters for various opening ratio

Table 7

 7

	Natural frequencies in water for various
	hole numbers
	N	In air [Hz]	In water [Hz]
	0	20.0	3.70
	1	18.7	4.30
	4	18.7	4.70
	9	19.0	5.30
	16	19.3	5.70
	25	18.7	5.70

Table 8

 8 Damping ratio and added mass in water for various opening ratios d/a Damping ratio Added mass[kg]

	0.0	0.10	1.17
	0.1	0.13	1.01
	0.3	0.18	0.88
	0.5	0.20	0.58
	0.6	0.23	0.45
	0.7	0.24	0.29
	N	Damping ratio Added mass [kg]
	0	0.10	1.17
	1	0.20	0.58
	4	0.30	0.45
	9	0.31	0.36
	16	0.29	0.32
	25	0.34	0.29

Table 9

 9 Damping ratio and added mass in water for various hole numbers