[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657435-105":59,"doc-detail-657435-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","block-diagram-modeling-of-second-order-systems-tutorial","Block Diagram Modeling of Second-Order Systems - Tutorial","","Block Diagram Modeling of Second-Order Systems explains how to represent a second-order differential equation with a Simulink block diagram. Using a single-degree-of-freedom mass-spring-dashpot system, it derives the equation of motion, rearranges it to form an explicit acceleration expression, and shows how gain blocks and a summing junction implement the bracketed term. It then demonstrates double integration to obtain displacement and uses feedback loops to connect ẋ and x within the model.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/block-diagram-modeling-of-second-order-systems-tutorial/657435/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/block-diagram-modeling-of-second-order-systems-tutorial/657435.png","ImageObject",300,407,{"name":92,"@type":93},"Damian","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What system and equation does the tutorial use as an example?","Question",{"text":112,"@type":113},"It uses a single-degree-of-freedom mass-spring-dashpot system and starts from the second-order equation of motion mx&& + cx& + kx = f(t).","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How is the term inside the brackets implemented in Simulink?",{"text":117,"@type":113},"The tutorial observes the bracketed signals f(t), x& , and x and uses gain blocks to multiply x& and x by their constants, then sums the modified inputs at a summing junction.",{"name":119,"@type":110,"acceptedAnswer":120},"How does the model obtain displacement x from the acceleration output x&&?",{"text":121,"@type":113},"It integrates the x&& output twice to obtain x, and creates feedback loops so the model uses ẋ and x consistently as outputs fed back into the integrator input ports.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657435,1792979043,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":52},137451208677,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Block Diagram Modeling of Second-Order Systems  \nINTRODUCTION  \nBlock diagrams are a method of describing the behavior of a dynamic system. In a block diagram, each discrete component, or block, represents part of the system. These blocks are connected together, representing how the “signal” flows between components. This can aid in understanding the interaction between various components, particularly for complex systems. Simulink allows block-diagram modeling of systems, and will be used for the examples in this tutorial. The concepts described here, however, are applicable to block diagrams in general.  \nIn this document, the basics of modeling second-order differential equations using block diagrams will be discussed. A single-degree-of-freedom mass-spring-dashpot system will be used as an example in the construction of the model. Other tutorials are available which contain more general information on second-order dynamic systems.  \nCONSTRUCTING THE MODEL  \nThe response of the mass-spring-dashpot system is described by a second-order differential equation known as the equation of motion,  \nmx&& + cx& + kx = f (t ) , (1)  \nwhere  \nm = the mass,  \nx = the displacement, x& = the velocity, x&& = the acceleration,  \nf(t) = the forcing function (input force),  \nc = the damping, and k = the spring constant.  \nFor the purposes of this tutorial,(1) will be rearranged to give  \nx&& = 1m [f(t ) − cx& − kx] . (2)  \nTo model this system, begin by observing the terms in the brackets. There are three input signal lines: f(t), x& , and x. Both x& and x are multiplied by constants, which is accomplished by using gain blocks. After each is appropriately modified, the three input signals are then summed together as shown in Fig.1 .  \nFig.1 . Summing the input signals.  \nThe block in the center of the diagram is referred to as a summing junction. Note that in the Simulink model labels,“xdot” and “xddot” refer to x& and x&& .  \nNext, it can be seen that the bracketed term [f(t ) − cx& − kx] is multiplied by the constant 1/m. Therefore, the output signal is passed through a gain block as shown in Fig.2 .  \nFig.2 . Applying a gain to the output.  \nThe output of this final gain block is x&& . The output of interest, however, is the displacement x. To obtain the x output signal, the x&& output signal is integrated twice, as shown in Fig.3 .  \nFig.3 . Integrating the output signal twice to obtain x.  \nAs can be seen in Fig.3, x& and x are both outputs from and inputs to the model. Therefore, two feedback loops are needed. A line is tapped off after the first integrator block and fed back into the open x& port, and a line is tapped off after the second integrator and fed back in as x. After adding a scope block (“Scope1”) to allow the output to be viewed in Simulink, and with some rearranging of blocks, the resulting model should look like Fig. 4. The block diagram is now complete.  \nFig. 4. The finished Simulink model for a second-order mass-spring-dashpot system.  \nEXAMPLE  \nNumerical values will now be entered into the Simulink model, shown in Fig. 5 below, so that it can be used to determine the step response of a second-order system.  \nFig. 5. Simulink model of second-order system.  \nConstruct the model as shown, and enter the values given in Table 2.  \nTable 2. Values used in Simulink model.  \n\n| Step block | Step time | 0 |\n| --- | --- | --- |\n|  | Initial value | 0 |\n|  | Final value | 1 |\n| Mass gain block | Gain | 1 |\n| Damping gain block | Gain | 10 |\n| Stiffness gain block | Gain | 500 |\n| Integration blocks (both) | Initial condition | 0 |\n\nWhen the result is viewed using the scope block, the result will appear as shown in Fig. 6.  \nFig. 6. Step response of second-order system, found using Simulink.","cbCaihcYOX0QHGhk","https://ap.wps.com/l/cbCaihcYOX0QHGhk","pdf",160543,"English","# Introduction\n# Constructing the Model\n# Example","[{\"question\":\"What system and equation does the tutorial use as an example?\",\"answer\":\"It uses a single-degree-of-freedom mass-spring-dashpot system and starts from the second-order equation of motion mx\\u0026\\u0026 + cx\\u0026 + kx = f(t).\"},{\"question\":\"How is the term inside the brackets implemented in Simulink?\",\"answer\":\"The tutorial observes the bracketed signals f(t), x\\u0026 , and x and uses gain blocks to multiply x\\u0026 and x by their constants, then sums the modified inputs at a summing junction.\"},{\"question\":\"How does the model obtain displacement x from the acceleration output x\\u0026\\u0026?\",\"answer\":\"It integrates the x\\u0026\\u0026 output twice to obtain x, and creates feedback loops so the model uses ẋ and x consistently as outputs fed back into the integrator input ports.\"}]","Block Diagram Modeling of Second-Order Systems - Tutorial | PDF"]