Contents lists available at ScienceDirect International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci Accurate Eigenvector-based generation and computational insights of Mindlin's plate modeshapes for twin frequencies Nabanita Datta ⁎ , Yogesh Verma Indian Institute of Technology, Kharagpur, India ARTICLE INFO Keywords: Plate vibration Modeshapes Eigenvectors Nodal patterns Trial functions ABSTRACT A semi-analytical approach to understand the manifestation of plate modeshapes associated with twin frequencies has been presented. Square Mindlin’s plate, clamped on all sides, has been considered here. It highlights the importance of efficacy of the beam-wise trial functions in an energy-based plate vibration analysis method, in terms of (a) accuracy, (b) orthogonality, (c) sense (plus/minus) and (d) interference. The inconsistency in the modeshapes of repeated frequencies, seen extensively in literature, has been attempted to be removed, through superior closed-form orthogonal set of Timoshenko admissible functions into the Rayleigh- Ritz method. The constructive/destructive interferences of the admissible functions, which are the products of the beam-wise modeshapes, give the final nodal patterns and the prominence of the anti-nodes. Also, the pairs of ‘very close’ but distinct frequencies, which were often considered as ‘numerical errors’, have been counter- intuitively justified through their Eigenvectors, which are either symmetric or skew-symmetric in the matrix form. Nodal patterns for CCCC plate modeshapes are accurately investigated; i.e. chess-board and diagonal nodal patterns. 1. Introduction Very little literature is available in the area of free vibration of plates which mathematically study plate modeshapes through the Eigenvectors of the Eigenvalue problem. Numerical and experimental attempts have been made to visualize the plate modeshapes, with limited success, in the first few frequencies only. Experimentally, modeshapes of the duplicate frequencies are not distinguishable, because they cannot co-exist simultaneously at a particular resonant excitation frequency of the vibration shaker. In real life, creating the exact classical clamped boundary condition can be a huge challenge, which leads to the invocation of random frequencies corresponding to plates with other undesired classical/non-classical boundary condi- tions. The premise of this work is as follows : • What causes inaccuracies in plate modeshapes generated by numer- ical techniques? • Why are plate modeshape often unpredictable in published litera- ture? • Why do the duplicate frequencies have dissimilar modeshapes by some numerical methods, while ‘very close’ pair of frequencies have unpredictable modeshapes in some other numerical methods? • Why do duplicate frequencies produce different plate modeshapes by numerical methods? This appears as a mathematical discrepancy. Repeated roots of an Eigenvalue problem have different Eigenvectors, but are they supposed to manifest different or duplicate modeshapes? • How important is the efficacy of the beam-wise trial functions in an energy-based plate vibration analysis method, in terms of (a) accuracy, (b) orthogonality, (c) sense ( ± ), (d) interference? • Only the frequency magnitude is no indication of the energy spread across the area of the plate. Frequency is the Eigenvalue, i.e. just a number, which does not indicate the spatial configuration of the vibration, which is important in dynamic stress analysis in various engineering applications. 2. Literature review and Work overview Ma and Huang [15] studied the modeshapes of a plate clamped on all sides (CCCC plate), and reported the first 12 frequencies, compared with FEA-based results. The experimental modeshape of the duplicate frequencies (Mode# 6) does not match the numerical ones. Mode# 11 shows a chess-board configuration, while the experiment shows a shaky diagonal nodal line. This questions the efficacy of either method, and necessitates the establishment of a robust approach to generate the accurate plate modeshapes. http://dx.doi.org/10.1016/j.ijmecsci.2017.01.044 Received 22 September 2016; Received in revised form 20 January 2017; Accepted 26 January 2017 ⁎ Corresponding author. E-mail address: nabanitadatta@gmail.com (N. Datta). International Journal of Mechanical Sciences 123 (2017) 64–73 Available online 29 January 2017 0020-7403/ © 2017 Published by Elsevier Ltd. MARK