Dynamic Matrix Research Articles

A computationally efficient method for dynamic response analyses of accessory drive systems using harmonic balance method together with alternating frequency/time domain technique

Serpentine belt accessory drive system (SBADS) is a typical coupled vibration system with rigid body vibrations and continuum vibrations. Vibration behaviors of a SBADS consist of rotational vibrations of pulleys and tensioner arm and transverse vibrations of belt spans. In modeling of a SBADS, transverse vibration of a belt span consisting of time and space coordinates is discretized into multiple degrees of freedom with only time variable. As a result, number of degrees of freedom of a SBADS is large, and computational efficiency of dynamic responses of a SBADS is low. To solve this issue, a calculation method for nonlinear dynamic characteristics of a SBADS is proposed based on the harmonic balance method with alternating frequency/time domain technique in the paper. An engine front end accessory drive system is taken as a study example, the coupling vibration model of the SBADS is established, and hysteretic behavior of the tensioner is considered in the model. In addition, a dimensionality reduction method for the dynamic matrix is proposed based on local nonlinearity of the model. A calculation method for nonlinear dynamic responses of the SBADS is subsequently developed by combining the harmonic balance method and alternating frequency/time domain technique. Nonlinear dynamic responses estimated by the proposed method are validated by comparing with numerical simulations. Finally, amplitude-frequency responses of the SBADS are calculated using arc-length continuation method, and influences of nonlinear tensioner on frequency domain responses and internal resonance phenomena of the SBADS are analyzed.

A fast polynomial-FE method for the vibration of the composite laminate quadrilateral plates and shells based on the segmentation strategy

The finite element method is widely applicable to various kinds of models. Nonetheless, as the size and frequency domain of the solving model increase, it encounters challenges related to high memory consumption and slow solution speed. To enhance the computational efficiency while preserving its existing advantages, this study incorporates the Jacobi polynomial with a segmentation strategy into the finite element method for addressing the vibration analysis of composite laminate quadrilateral flat and curved plates. In this method, the dynamic stiffness matrix of a finite element is multiplied on both sides by a matrix composed of Jacobi polynomial basis functions, thereby transforming the expression form of the dynamic stiffness matrix from FE form to meshless form and reducing its size. Additionally, the segmentation strategy is introduced to divide the model into multiple blocks, enhancing convergence and calculation efficiency by altering the sparsity pattern of the dynamic stiffness matrix. Convergence research, validation studies, limitations studies, and parametric studies are conducted. The method demonstrates evident advantages in computational efficiency for random surface quadrilateral models requiring dense mesh description while maintaining a certain level of accuracy.

Study on Multivariable Dynamic Matrix Control for a Novel Solar Hybrid STIGT System

To construct a clean and efficient energy system, advanced solar thermal power generation technology is developed, i.e., a solar hybrid STIGT (Steam Injected Gas Turbine) system with near zero water supply. Such a system is conducive to the efficient use of solar energy and water resources, and to improvement of the performance of the overall system. Given that the strong correlation between multiple-input and multiple-output of the new system, the MDMC (Multivariable Dynamic Matrix Control) method is proposed as an alternative to a PID (Proportional-Integral-Derivative) controller to meet requirements in achieving better control characteristics for a complex power system. First, based on MATLAB/Simulink, a dynamic model of the novel system is established. Then it is validated by both experimental and literature data, yielding an error no more than 5%. Subsequently, simulation results demonstrate that the overshoot of output power on MDMC is 1.2%, lower than the 3.4% observed with the PID controller. This improvement in stability, along with a reduction in settling time and peak time by over 50%, highlights the excellent potential of the MDMC in controlling overshoot and settling time in the novel system, while providing enhanced stability, rapidity, and accuracy in the regulation and control of distribution networks.

Fast-relaxing hydrogels with reversibly tunable mechanics for dynamic cancer cell culture

The mechanics of the tumor microenvironment (TME) significantly impact disease progression and the efficacy of anti-cancer therapeutics. While it is recognized that advanced in vitro cancer models will benefit cancer research, none of the current engineered extracellular matrices (ECM) adequately recapitulate the highly dynamic TME. Through integrating reversible boronate-ester bonding and dithiolane ring-opening polymerization, we fabricated synthetic polymer hydrogels with tumor-mimetic fast relaxation and reversibly tunable elastic moduli. Importantly, the crosslinking and dynamic stiffening of matrix mechanics were achieved in the absence of a photoinitiator, often the source of cytotoxicity. Central to this strategy was Poly(PEGA-co-LAA-co-AAPBA) (PELA), a highly defined polymer synthesized by reversible addition-fragmentation chain transfer (RAFT) polymerization. PELA contains dithiolane for initiator-free gel crosslinking, stiffening, and softening, as well as boronic acid for complexation with diol-containing polymers to give rise to tunable viscoelasticity. PELA hydrogels were highly cytocompatible for dynamic culture of patient-derived pancreatic cancer cells. It was found that the fast-relaxing matrix induced mesenchymal phenotype of cancer cells, and dynamic matrix stiffening restricted tumor spheroid growth. Moreover, this new dynamic viscoelastic hydrogel system permitted sequential stiffening and softening to mimic the physical changes of TME.

Evolution of Crop Planting Structure in Traditional Agricultural Areas and Its Influence Factors: A Case Study in Alar Reclamation

This research provides a comprehensive analysis of the spatiotemporal evolution of the regional cropping structure and its influencing factors. Using Landsat satellite images, field surveys, and yearbook data, we developed a planting structure extraction model employing the classification regression tree algorithm to obtain data on the major crop cultivation and structural characteristics of Alar reclamation from 1990 to 2023. A dynamic model and transfer matrix were used to analyze temporal changes, and a centroid migration model was used to study spatial changes in the cropping structure. Nonparametric mutation tests and through-traffic coefficient analysis were utilized to quantify the main driving factors influencing the cropping structure. During the period of 1990–2023, the cotton area in the Alar reclamation region expanded by 722.08 km2, while the jujube exhibited an initial increase followed by a decrease in the same period. The primary reasons are linked to the cost of purchase, agricultural mechanization, and crop compatibility. In the Alar reclamation area, cotton, chili, and jujube are the primary cultivated crops. Cotton is mainly grown on the southern side of the Tarim River, while chili cultivation is concentrated on the northern bank of the river. Over the years, there has been a noticeable spatial complementarity in the distribution and density of rice and cotton crops in this region. In the Alar reclamation, the main factors influencing the change in cultivated land area are cotton price, agricultural machinery gross power, and population. Consequently, implementing measures such as providing planting subsidies and other policy incentives to enhance planting income can effectively stimulate farmers’ willingness to engage in planting activities.

Research on roller bearing fault diagnosis method based on flexible dynamic adjustable strategy under data imbalance

In engineering practice, the collection of equipment vibration signals is prone to interference from the external environment, resulting in abnormal data and imbalanced data in different states. Traditional support vector machine, support matrix machine and other methods have advantages in balancing sample classification, but have limitations in obtaining low rank information, making it difficult to perform classification tasks under data imbalance. Therefore, a novel classification method that targets matrices as the input, called flexible dynamic matrix machine (FDMM), is proposed in this paper. First, FDMM establishes a regularization term using a flexible low-rank operator and sparse constrain, which can better take into account matrix structure information. Then, the upper bound of the loss function is truncated, reducing the impact of the loss on the construction of the decision hyperplane. Finally, the recognition performance of imbalanced data is improved by adjusting the game values of different categories of samples through dynamic adjustment function. Experimental results demonstrate that superior classification accuracy and generalization performance can be achieved with the FDMM method when applied to two roller bearing datasets.

Soil metabolomics - current challenges and future perspectives

Soil is an extremely complex and dynamic matrix, in part, due to the wide diversity of organisms living within it. Soil organic matter (SOM) is the fundamental substrate on which the delivery of ecosystem services depends, providing the metabolic fuel to drive soil function. As such, studying the soil metabolome (the diversity and concentration of low molecular weight metabolites), as a subset of SOM, holds the potential to greatly expand our understanding of the behaviour, fate, interaction and functional significance of small organic molecules in soil. Encompassing a wide range of chemical classes (including amino acids, peptides, lipids and carbohydrates) and a large number of individual molecules (ca. n = 105 to 106), the metabolome is a resultant (indirect) output of several layers of a biological hierarchy, namely the metagenome, metatranscriptome and metaproteome. As such, it may also provide support and validation for these “multi-omics” datasets. We present a case for the increased use of untargeted metabolomics in soil biochemistry, particularly for furthering our fundamental understanding of the functions driving SOM composition and biogeochemical cycling. Further, we discuss the scale of the challenge in terms of metabolite extraction, analysis and interpretation in complex plant-soil-microbial systems. Lastly, we highlight key knowledge gaps which currently limit our use of metabolomic approaches to better understand soil processes, including: (i) interpretation of large untargeted metabolomic datasets; (ii) the source, emission and fate of soil-derived volatile organic compounds (VOCs), (iii) assessing temporal fluxes of metabolites, and (iv) monitoring ecological interactions in the rhizosphere. While the application of metabolomics in ecosystem science is still in its relative infancy, its importance in understanding the biochemical system in relation to regulation, management and underpinning the delivery of ecosystem services is key to further elucidating the complex links between organisms, as well as the fundamental ability of the biological community to process and cycle key nutrients.

Tailoring the Degradation Time of Polycationic PEG-Based Hydrogels toward Dynamic Cell Culture Matrices.

Poly(ethylene glycol)-based (PEG) hydrogels provide an ideal platform to obtain well-defined and tailor-made cell culture matrices to enhance in vitro cell culture conditions, although cell adhesion is often challenging when the cells are cultivated on the substrate surface. We herein demonstrate two approaches for the synthesis of polycationic PEG-based hydrogels which were modified to enhance cell-matrix interactions, to improve two-dimensional (2D) cell culture, and catalyze hydrolytic degradation. While the utilization of N,N-(bisacryloxyethyl) amine (BAA) as cross-linker for in situ gelation provides degradable scaffolds for dynamic cell culture, the incorporation of short segments of poly(N-(3-(dimethylamino)propyl)acrylamide) (PDMAPAam) provides high local cationic charge density leading to PEG-based hydrogels with high selectivity for fibroblastic cell lines. The adsorption of transforming growth factor (TGF-β) into the hydrogels induced stimulation of fibrosis and thus the formation of collagen as a natural ECM compound. With this, these dynamic hydrogels enhance in vitro cell culture by providing a well-defined, artificial, and degradable matrix that stimulates cells to produce their own natural scaffold within a defined time frame.

Sustainability evaluation of urban large-scale infrastructure construction based on dynamic fuzzy cognitive map

Enhancing the sustainability of large-scale infrastructure construction is the foundation and trend of urban sustainable development and environmental protection. This paper proposes a perception and evaluation method based on Dynamic Fuzzy Cognitive Mapping (D-FCM) to address the dynamics, uncertainties, and implementation difficulties of sustainability evaluation in large-scale infrastructure construction. Using Hebbian rules to learn the experience and knowledge of experts, a dynamic evolution correlation matrix about evaluation indicators is obtained, and a D-FCM dynamic system model is established to obtain the sustainability evaluation results of large-scale infrastructure construction. The comprehensive sustainable evaluation value (K) of the selected subway project case is 0.934, which belongs to the V level in the sustainable level classification standard and has a good trend in sustainable development. Social public participation (S1), light pollution prevention (H1), wastewater discharge compliance rate (H2), soil pollution prevention (H3), and soil and water conservation and utilization (Z2) are the most significant and sensitive indicators for the sustainable development of large-scale infrastructure construction. Based on the interaction of influencing factors in the D-FCM model, an economically efficient control measures and decision-making plan can be proposed to improve the sustainability of large-scale infrastructure construction projects.

Mechanically Robust Hemostatic Hydrogel Membranes with Programmable Strain-Adaptive Microdomain Entanglement for Wound Treatment in Dynamic Tissues.

Supramolecular hydrogels emerge as a promising paradigm for sutureless wound management. However, their translation is still challenged by the insufficient mechanical robustness in the context of complex wounds in dynamic tissues. Herein, we report a tissue-adhesive supramolecular hydrogel membrane based on biocompatible precursors for dressing wounds in highly dynamic tissues, featuring robust mechanical resilience through programmable strain-adaptive entanglement among microdomains. Specifically, the hydrogels are synthesized by incorporating a long-chain polyurethane segment into a Schiff base-ligated short-chain oxidized cellulose/quaternized chitosan network via acylhydrazone bonding, which readily establishes interpenetrating entangled microdomains in dynamic cross-linked hydrogel matrices to enhance their tear and fatigue resistance against extreme mechanical stresses. After being placed onto dynamic tissues, the hydrogel dressing could efficiently absorb blood to achieve rapid hemostasis. Moreover, metal ions released from ruptured erythrocytes could be scavenged by the Schiff base linkers to form additional ionic bonds, which would trigger the cross-linking of the short-chain components and establish abundant crystalline microdomains, eventually leading to the in situ stiffening of the hydrogels to endure heavy mechanical loads. Benefiting from its hemostatic capacity and strain adaptable mechanical performance, this hydrogel wound dressing shows promise for the clinical management of various traumatic wounds.

Bioinspired Dynamic Matrix Based on Developable Structure of MXene‐Cellulose Nanofibers (CNF) Soft Actuators

AbstractInspired by natural organisms, soft actuators can convert environmental stimuli into mechanical deformation, making them indispensable for applications in a variety of fields such as soft robotics. MXene, displaying exceptional attributes in conductivity, thermal efficiency, good dispersibility etc., has emerged as a preferred material for high‐performance thermal actuators. However, single actuators struggle to achieve complex tasks realized by intelligent robotic systems. Herein, a bioinspired dynamic matrix is presented utilizing the developable structure of MXene‐cellulose nanofibers (CNF) soft actuators. The inclusion of CNF considerably bolsters the mechanical properties of MXene. The MXene‐CNF film possessed a higher working strain range (≈14%) than pure MXene film (≈2%). The designed developable structures complete the actuating movements. The sensing layer integration with the actuating layer led to an extremely low touch detection limit (0.3 kPa) and expedited actuating‐feedback due to the interaction between contact charging and electrostatic induction. A responsive dynamic matrix containing 3 × 3 soft actuators, completed with a close‐looped sensing‐feedback function, is developed similar to the behavior of the mimosa plant. This mimosa‐inspired dynamic matrix is capable of identifying the source of touch stimuli and providing immediate feedback. This research broadens the potential for enhancing adaptability and intelligence of soft robotic system.

A Trustworthy and Explainable Framework for Benchmarking Hybrid Deep Learning Models Based on Chest X-Ray Analysis in CAD Systems

Evaluating the trustworthiness of deep learning-based computer-aided diagnosis (CAD) systems is challenging. There is a need to optimize trust and performance in model selection. A wide range of models based on evaluation metrics can make it challenging to choose the best one, especially for complex multi-criteria decision-making problems. In the case of COVID-19 diagnosis, using both physicians’ evaluation and AI techniques to establish trust is essential. In this study, 1551 chest X-rays were analyzed using deep transfer learning (DTL) with six models and four SVM kernels. This resulted in 24 hybrid DTL–SVM models. Seven metrics were evaluated using fuzzy multiple-criteria decision-making (MCDM), which uses a dynamic decision matrix to select the best model. This matrix incorporates fuzzy-weighted zero-inconsistency (FWZIC) for weight coefficients and Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) for benchmarking. The Grad-CAM technique compared the best model with 16 images, ensuring explainability. Top-performing models were identified, including SqueezeNet-SVM linear, VGG19-SVM linear, and VGG19-SVM. Sensitivity analysis was used to quantify the impact of changing weighted criteria values. A physician expert validated fuzzy MCDM through Grad-CAM analysis, a new aspect of this study. The framework presented in this study was benchmarked against seven other studies and achieved a perfect score in four crucial areas. Trustworthiness is essential for CAD systems, and this study effectively addresses trust and performance challenges in AI model-based CAD systems. The study systematically evaluated the requirements for trustworthiness, including accountability, fairness, robustness, accuracy, and reproducibility, and the results supported this.

Dynamic matrix factorization-based collaborative filtering in movie recommendation services

Movies are a primary source of entertainment, but finding specific content can be challenging given the exponentially increasing number of movies produced each year. Recommendation systems are extremely useful for solving this problem. While various approaches exist, Collaborative Filtering (CF) is the most straightforward. CF leverages user input and historical preferences to determine user similarity and suggest movies. Matrix Factorization (MF) is one of the most popular Collaborative Filtering (CF) techniques. It maps users and items into a joint latent space, using a vector of latent features to represent each user or item. However, traditional MF techniques are static, while user cognition and product variety are constantly evolving. As a result, traditional MF approaches struggle to accommodate the dynamic nature of user-item interactions. To address this challenge, we propose a Dynamic Matrix Factorization CF model for movie recommendation systems (DMF-CF) that considers the dynamic changes in user interactions. To validate our approach, we conducted evaluations using the standard MovieLens dataset and compared it to state-of-the-art models. Our preliminary findings highlight the substantial benefits of DMF-CF, which outperforms recent models on the MovieLens-100K and MovieLens-1M datasets in terms of Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) metrics.

A novel dynamic spatio-temporal graph convolutional network for wind speed interval prediction

It is crucial to predict the wind speed for the utilization of renewable wind energy and the operation of transmission lines with increased capacity. The intermittency and stochastic fluctuations of wind speed pose a significant challenge for the high-quality wind speed prediction, and a novel wind speed interval prediction (WSIP) model is constructed in this study by employing the residual estimation (RE)-oriented dynamic spatio-temporal graph convolutional network (DSTGCN) approach. Firstly, a dynamic adjacency matrix is designed to obtain time-varying global spatial weight allocations among each wind speed node. Then, the spatio-temporal features are extracted by using gated recurrent units (GRUs) and GCNs to construct the wind speed graph networks. Moreover, the RE-oriented strategy incorporating the pinball loss is designed to provide a guidance the parameter training of the constructed model, thus eliminating the quantile crossings problem. As a result, the deterministic point prediction of the wind speed is expanded to the quantile-based probabilistic interval prediction. Finally, the experimental results are presented to demonstrate the validity and superiority of proposed scheme in both qualitative and quantitative performance.

Dynamic Quantized Consensus Under DoS Attacks: Towards a Tight Zooming-Out Factor

This paper deals with dynamic quantized consensus of dynamical agents in a general form under packet losses induced by Denial-of-Service (DoS) attacks. The communication channel has limited bandwidth and hence the transmitted signals over the network are subject to quantization. To deal with agent's output, an observer is implemented at each node. The state of the observer is quantized by a finite-level quantizer and then transmitted over the network. To solve the problem of quantizer overflow under malicious packet losses, a zooming-in and out dynamic quantization mechanism is designed. By the new quantized controller proposed in the paper, the zooming-out factor is lower bounded by the spectral radius of the agent's dynamic matrix. A sufficient condition of quantization range is provided under which the finite-level quantizer is free of overflow. A sufficient condition of tolerable DoS attacks for achieving consensus is also provided. At last, we study scalar dynamical agents as a special case and further tighten the zooming-out factor to a value smaller than the agent's dynamic parameter. Under such a zooming-out factor, it is possible to recover the level of tolerable DoS attacks to that of unquantized consensus, and the quantizer is free of overflow.

Exergy-based control strategy design and dynamic performance enhancement for parabolic trough solar receiver-reactor of methanol decomposition reaction

The parabolic trough solar receiver-reactors of methanol decomposition reaction (PTSRR-MDR) enable efficient hydrogen production with solar energy at users’ end. Solar fluctuations affect the solar-to-chemical efficiency, hydrogen yield and system stability, indicating the necessity for an appropriate control strategy. In this study, dynamic and thermodynamic models of PTSRR-MDR with Cu/ZnO/Al2O3 as catalyst were developed. Results of thermodynamic analysis show that an optimal flow rate exists for methanol to achieve the maximum solar-to-chemical exergy efficiency of the PTSRR-MDR. The temperature increases sharply and exceeds the catalyst sintering temperature when the methanol flow rate decreases under its optimal value. Then, control objectives to achieve the maximum efficiency are obtained with a temperature margin of 3 °C to guarantee the PTSRR-MDR safety. Control strategies based on Dynamic Matrix Control (DMC), Forward-Feedback Control (FFC) and Ramp Control (RC) were evaluated under dynamic simulations. Results show that strategies based on DMC, FFC, and RC increase the solar-to-chemical exergy efficiency from 14.76% to 16.31%, 16.28%, and 15.65%, respectively. The Integral Squared Error (ISE) of methanol conversion rate under three methods are 1.46, 0.32 and 5.28, respectively, indicating that strategy based on FFC is considered most suitable for PTSRR-MDR operation control.

High‐Performance Te Nanowires/MoS2/Polyimine Nanocomposite‐Based Self‐Healable, Recyclable and Screen‐Printable Flexible Photodetector for Image Sensing

AbstractIntrinsically flexible photodetectors are compelling building blocks for next‐generation wearable optoelectronic systems owing to their distinctive advantages of reliable structural durability and versatile scalability for large‐scale production. However, their practical applications are still impeded by the inferior photodetection performance, irreversible device failure after breakage, and serious e‐waste accumulation after service life. Herein, a high‐performance intrinsically flexible, mechanically durable, self‐healable, closed‐loop recyclable, and screen‐printable Te NWs/MoS2 nanosheets/polyimine nanocomposite‐based photodetector are designed by engineering‐ordered‐bridged 1D/2D carrier percolation “fast lanes” in dynamic covalent polyimine matrix via a flow‐designed solution‐shearing method. Such a design provides a sixfold, 20.1‐fold, and 6.9‐fold enhancement in carrier mobility, responsivity (11.68 mA W−1), and detectivity (1.145 × 1010 Jones), respectively, as well as stable photoresponse over eight months or after 50 000 bending‐flattening times. Meanwhile, this photodetector presents excellent self‐healing efficiency and repeatable recyclability for device reconfiguration. Furthermore, these merits can be fully integrated onto textile by assembling nacre‐like Te NWs/MoS2/polyimine nanocomposite coatings on textiles via screen‐printing processes, enabling programmable patterning of photodetection arrays for large‐area image sensing. This work provides a viable approach for the design of shape‐tunable optoelectronics with reliable mechanical durability and customizable functionalities, demonstrating the tremendous potential for large‐scale applications in wearable optoelectronic systems.

Dynamic stiffness method and CUF-based component-wise theories applied to free vibration analysis of solid beams, thin-walled structures and reinforced panels

In this paper, an exact dynamic stiffness formulation using higher-order theories with displacement variables only is presented and subsequently used to investigate the free vibration characteristics of solid beams, thin-walled structures and reinforced panel structures. In essence, higher-order displacement fields are developed by using the Carrera unified formulation (CUF), and by discretizing the cross-section kinematics with bilinear, cubic and fourth-order Lagrange polynomials. In particular, the component-wise (CW) approach based on Lagrange expansion is applied in which the solid part and thin-walled part are considered as two independent components that can be assembled. The principle of virtual displacements is used to derive the governing differential equations and the associated natural boundary conditions. An exact dynamic stiffness matrix is then developed by relating the amplitudes of harmonically varying loads to those of the responses. The explicit terms of the dynamic stiffness matrices are also presented. The Wittrick–Williams algorithm of the dynamic stiffness method (DSM) is applied with the explicit expressions of the J0 count for beam elements under all above support conditions. In return, there is no need to refine the element in the DSM, and thus, it becomes immensely efficient. The accuracy and efficiency of the proposed methodology are extensively assessed for different solid beams, thin-walled structures and reinforced panels and the results are compared with those appearing in published literature and also checked by 3D finite element (FE) solutions.

Dynamic Matrix Recovery

Matrix recovery from sparse observations is an extensively studied topic emerging in various applications, such as recommendation system and signal processing, which includes the matrix completion and compressed sensing models as special cases. In this article, we propose a general framework for dynamic matrix recovery of low-rank matrices that evolve smoothly over time. We start from the setting that the observations are independent across time, then extend to the setting that both the design matrix and noise possess certain temporal correlation via modified concentration inequalities. By pooling neighboring observations, we obtain sharp estimation error bounds of both settings, showing the influence of the underlying smoothness, the dependence and effective samples. We propose a dynamic fast iterative shrinkage-thresholding algorithm that is computationally efficient, and characterize the interplay between algorithmic and statistical convergence. Simulated and real data examples are provided to support such findings. Supplementary materials for this article are available online.

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.