The mass-spring-damper system provides a nice example to\r\nillustrate these three steps. Let\u2019s look at the equation for this system:<\/p>\r\n\r\n
![\"Spring](\"https:\/\/blogs.mathworks.com\/images\/seth\/2008Q2\/smdEquation.png\"){kind=small}
<\/p>\r\n\r\n
The position of the mass is Express the system as first order derivatives<\/strong><\/p>\r\n\r\n To rewrite this as a system of first order derivatives, I\r\nwant to substitute And this is rewritten at two first derivatives:<\/p>\r\n\r\n Velocity and position are the states of my system. When\r\nthinking about ODEs, states equal integrator blocks.<\/p>\r\n\r\n Add one integrator per state, label the input and output<\/strong><\/p>\r\n\r\n I always make a point to write the equations as an\r\nannotation on my diagram. I refer to this as I add blocks to the canvas. Here\r\nare the two integrators for the mass-spring-damper system.<\/p>\r\n\r\n I draw signals from the ports and label inputs as the\r\nderivative ( Connect the terms to form the system<\/strong><\/p>\r\n\r\n The first connection is easy, To implement the second equation, I add gains and sums to\r\nthe diagram and link up the terms.<\/p>\r\n\r\n The final step, initial conditions<\/strong><\/p>\r\n\r\n Modeling differential equations require initial conditions\r\nfor the states in order to simulate. The initial states are set in the\r\nintegrator blocks. Think of these as the initial value for v and x at time 0. \r\nThe ODE solvers compute the derivatives at time zero using these initial\r\nconditions and then propagate the system forward in time. I used an annotation\r\nto record these initial conditions, v0 = 0, and x0 = 10.<\/p>\r\n\r\n Simulating the model for 50 seconds produces the following\r\ntrace for x (blue) and v (red).<\/p>\r\n\r\n Over time, I have become more comfortable in converting from\r\nequations to models, and I do not always rewrite the states. I think that\r\nfundamentally I still follow the same process:<\/p>\r\n\r\n Now it\u2019s your turn<\/strong><\/p>\r\n\r\n Is this the same process you learned when you started using\r\nSimulink? Do you have any special tips on how to draw ODEs? Share your ideas\r\nand leave a\r\ncomment here<\/a>.<\/p>\r\n","protected":false},"excerpt":{"rendered":" I remember while learning Simulink, drawing ordinary\r\ndifferential equations was one of the early challenges. Eventually I\r\ndiscovered a few steps that make it easier. First, rewrite the equations... read more >><\/a><\/p>","protected":false},"author":40,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[30,29],"tags":[32,441,31],"_links":{"self":[{"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/posts\/15"}],"collection":[{"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/users\/40"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/comments?post=15"}],"version-history":[{"count":0,"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/posts\/15\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/media?parent=15"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/categories?post=15"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.mathworks.com\/simulink\/wp-json\/wp\/v2\/tags?post=15"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![](\"https:\/\/blogs.mathworks.com\/images\/seth\/2008Q2\/x.png\"){kind=small}
<\/span>,\r\nthe velocity is <\/span>, and the acceleration is
<\/span>. <\/p>\r\n\r\n
<\/span> for
<\/span>, and
<\/span> for
<\/span>. Then I can identify my two\r\nstates as position
<\/span> and velocity
<\/span>. The equation becomes<\/p>\r\n\r\n
\r\n<\/p>\r\n\r\n
<\/span>), and the output is the state\r\nvariable.<\/p>\r\n\r\n
<\/span>, so I connect the output of the\r\nvelocity integrator to the input of the position integrator. When this\r\nhappens, aligning the integrators in the diagram shows that you have a second\r\norder system.<\/p>\r\n\r\n
<\/p>\r\n\r\n
<\/p>\r\n\r\n
<\/p>\r\n\r\n
\r\n