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# Double Pendulum Equations Of Motion Small Angle

(See the diagram below). The coupling will be assumed weak. For small angles of oscillation, we take the Lagrangian to be. But equation (6) is not easy to use. The full range of motion of the double pendulum is described by two complicated equations. The Euler-Lagrange Equations, a fundamental result of the calculus of variations, combined with the Principle of Least Action and the Lagrange D’Alembert Principle are used to describe the equations of motion for a cart and double-pendulum system in terms of the kinetic and potential energy of the system, which is in turn described in terms of the positions and velocities of the cart and two pendulum bobs. In Stickel (2009), the Lagrangian is representation system of motion and can be used when system is conservative. However, with increasing amplitude, the linear equation ceases to be valid. Pendulum (mathematics)'s wiki: The mathematics of pendulums are in general quite complicated. Employing state-space design techniques, we are more readily able to address a multi-output system. However, since these equations have no analytical solution. KE i + PE i = KE f +PE f [½mv 2 + mgL(1-COSq) ] i = [½mv 2 + mgL(1-COSq) ] f. To do this we go through the same steps. The angular equation of motion of the pendulum is simply. For the double pendulum. Summary: The equation of motion is d 2 θ dt2 + g L sinθ = 0 Go to derivation. In our case, the inverted pendulum system is single-input, multi-output (SIMO). equations for the double pendulum motion analysis are given. Principle are used to describe the equations of motion for a cart and double-pendulum system in terms of the kinetic and potential energy of the system, which is in turn described in terms of the positions and velocities of the cart and two pendulum bobs. Stopped at the second time derivatives of the constraint equations. In this paper, a novel online motion planning method for double-pendulum overhead cranes is proposed. Forced pendulum that can display chaotic motion E m = 1 2 ˙2 g l (cos + µ cos( ⌫t)). (a) The Lagrangian for a spherical pendulum Let’s assume that the mass is on \bottom half" of the sphere, so that the mass has a Cartesian coordinate z = lcos. 005 (part 2 of 2) 10 points Based on the equation of the motion given in the previous question, the period of this pendulum in the small angle approximation is given by Correct answer: 3. For example, the appendix of this document lists the three coupled second-order differential equations that govern the motion of the a triple pendulum; here's a screenshot of just the first of. The equation of motion is not changed from that of a simple pendulum, but the energy is not constant. In fact, White says, they are too complicated to be of much help in a study of the golf swing. Figure 3 – Double Pendulum. The motion of the double pendulum is completely described by its (complicated) equations of motion, but its behavior depends sensitively to small changes in initial conditions. lum, and is the displaced angle. If the oscillation of driven torsion pendulum is confined to moderate angles (smaller then say 30°) the motion may be accurately modeled by the Duffing equation [8]. If I proceed adding a -kx 1 to F x 2 and solving for the equations of motion using the small angle approximation I get Help with coupled spring and pendulum system S. A double pendulum is a piece of mass attached to another piece of mass via a piece of rigid wire or string. A similar mild steel rod is xed to the lower end of either upper limb from which the lower limbs hang. It is a simple physical system that exhibits mathematical chaos. It makes a pretty curve called a Lissajous curve. Define a phase space, and determine an equation of the form for the double pendulum shown in Figure 13. and the relevant moment of inertia is that about the point of suspension. According to your expression it changes linearly with time, although the equation is harmonic: θ = θ 0 cos(ωt) where ω = g/L, L is the length of the pendulum and g the acceleration due to gravity. Jeremiah Ruesch Department of Physics California State University, Chico Abstract: As many undergraduates are aware, the first nonlinear system often encountered is the simple plane pendulum when the small angle approximation in the equations of motion is not assumed. The reason why it applies to so many situations is the following. The graph on the left show one angle as a function of the other evolving in time. Simple gravity pendulum. This occurs because the pendulum moves back and forth (first in one direction, then the other) for one complete period; giving two peaks in the Y acceleration graph per swing of the pendulum's motion. This is because the motion is determined by simple sine and cosine functions. on the system have to be known and be able to be expressed. For this purpose, the whole walking motion can be simulated as an inverted pendulum, where one-leg locomotion with a stopper at the lower joint corresponds to a self-impact double pendulum; controlling of which is a benchmark problem in biomedical and control engineering. Double Pendulum Java Application Physics Background. t/ D 2sin 1 k sn r g L. small-angle pendulum, a torsion oscillator, certain electrical circuits, sound vibrations, molecular vibrations, and countless other setups. Photogate folder, double click on the Pendulum Timer file. From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. From Wikipedia, the free encyclopedia. Oscillations we obtain the equation of motion for a mass on a spring. ~ (18) -B, -mZg = m2aE,y (19) 1, cos o2 1, sin0, 1 42 + By2 = ~m21~0L2. Formulate a Lagrangian to describe the dynamics. , we start with the assumption that. A double pendulum released from a small initial angle behaves similarly to the single pendulum. Photogate folder, double click on the Pendulum Timer file. For example: MathWorld, Wikipedia, and myphysicslab. A double pendulum consists of two simple pendulums of lengths l1 and l2 and masses m1 and m2 , with the cord of one pendulum attached to the bob of another pendulum whose cord is fixed to a pivot, Fig. Use the generalized coordinates 2 and N as shown. the equation for $\theta$). Equations of motion for a planar simple double pendulum March 17, 2016 / 2 Comments To provide some background information for my N-link pendulum project, I’ve broken the methodology for solving the equations of motion (EOM) for a simple double pendulum into a separate post. The pendulum was an instrument of torture and execution claimed to have been used by the Spanish Inquisition as recently as the early 19th Century. The intent of the experiment is to investigate the motion of one particular kind called the bi lar pendulum. A question on Electrical Engineering SE explains the driving circuit. We study the stabilization of a double pendulum mounted on a movable cart. with notebook. The double inverted pendulum is shown in Fig 1. Double pendulum governing equation The double pendulum is the simplest mechanical apparatus that exhibits a range of dynamic responses from periodic oscillations to chaotic rotations. 2 − ( 1 + 2) 1. Acceleration due to gravity is g. The specific nature of pendulum motion, and simple harmonic motion in general, is that in the course of. If desired, make the lower joint of the demonstration double pendulum immobile with tongue depressors and. which the double pendulum motion can be described mathematically (e. $$Using this substitution in the Lagrangian, and expanding to quadratic order in \epsilon will produce the correct linearlized equations of motion. Double pendulum governing equation The double pendulum is the simplest mechanical apparatus that exhibits a range of dynamic responses from periodic oscillations to chaotic rotations. which is known as Christiaan Huygens's law for the period. [mex132] Plane pendulum I Derive the equation of motion φ¨+(g/ℓ)sinφ = 0 for the (generalized) angular coordinate φ within the framework of Newtonian mechanics by proceeding as follows: (a) State the two equations of motion for the Cartesian coordinates x,y in terms of the known applied force mg and the unknown force of constraint. The double pendulum is a dynamic system that exhibits sensitive dependence upon initial conditions. Large-angle oscillations are not simple harmonic because the restoring torque is not proportional to the angular displacement. Use Lagrange’s equations to derive the equation of motion for a single-link pendulum consisting of a uniform slender rod of mass. (a) The Lagrangian for a spherical pendulum Let’s assume that the mass is on \bottom half" of the sphere, so that the mass has a Cartesian coordinate z = lcos. The full range of motion of the double pendulum is described by two complicated equations. For small deviations from equilibrium, these oscillations are harmonic and can be described by sine or cosine function. We assume that this model will only be used when the pendulums have a relatively small angular displacement. d2 dt2 ˇ g L (4) 2 + g L = 0 = p g L i in the form of a bi using the complex roots case for solving second-order equations y(t. The dynamics of the double pendulum are chaotic and complex, as illustrated below. Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, the strength of gravity, and the amplitude of the swing. Everywhere you look this problem is always simplified, the reasons for this are clear, before the advent of computers and especially software like Maple there was no way of solving the equations of motion and describing the time evolution of the motion. A simple pendulum consists of a mass m hanging from a string of length L and fixed at a pivot point P. Find the period of the pendulum. In fact, the equations of motion can be written more easily if several coordinate systems are employed. The nature of the Pendulum Program: The Pendulum file is a program that uses the photogate to implement an event timer specific to pendulum motion. However, since these equations have no analytical solution. Hamiltonian equations of motion. Obviously, simple pendulums do not exist. Like, maybe this is only 20 degrees or less, that pendulum would be described really well by this equation because it would be extremely close to being a simple harmonic oscillator. From Figure. Our aim is to solve eqs (12) and (16) using finite differences , but first we offer an example with some common approximations. Thus, s = Lθ, where θ must be measured in radians. Normal Modes. Two degree of freedom system. Note: the EOM derivation is left as an exercise to the reader. When displaced to an initial angle and released, the pendulum will swing back and forth with periodic motion. Disclosed are various embodiments of methods and systems related to double-pendulum crane control. The double pendulum is a pendulum hanging from a pendulum. Handed out assignment 1, this homework is due next monday. exhibit sensitive dependence toinitialconditions. Related End-of-Chapter Exercises: 57, 58. Forced pendulum that can display chaotic motion E m = 1 2 ˙2 g l (cos + µ cos( ⌫t)). ~o~ a do1;1~le pendulum (released from a large initial angle), It IS pOSitiVe. nc i i i i d T T V ( ) Q , i 1,2,3,. In Stickel (2009), the Lagrangian is representation system of motion and can be used when system is conservative. A driven single pendulum and a double pendulum (free or driven) are examples of chaotic systems. A double pendulum released from a small initial angle behaves similarly to the single pendulum. 10) we get the differential equations of the angle. It can be shown mathematically that the position-time graph of a pendulum is sinusoidal, with a period that depends on the pendulum’s length. Period of Oscillation of a Nonlinear Pendulum. Phys 7221 Hwk #9: Small Oscillations Gabriela Gonz´alez December 5, 2006 Prob 6-4: Double Pendulum We follow the conventions for angles in Figure 1. A double pendulum consists of two simple pendulums of lengths l1 and l2 and masses m1 and m2 , with the cord of one pendulum attached to the bob of another pendulum whose cord is fixed to a pivot, Fig. INTRODUCTION. General Discussion The time period for small amplitudes For very small amplitudes, the motion of the simple pendulum may be approximated by simple harmonic motion. MATLAB/Simulink - Simple Pendulum: Pendulum's equation of motion done in Simulink using ode45 solver. The figure shows tangential and radial components of gravitational force on the pendulum bob. By applying Newton's secont law for rotational systems, the equation of motion for the pendulum may be obtained , and rearranged as. In a simple double pendulum, the mass is distributed through the length, hence the angle between two arms is used to calculate moment of Inertia. In this paper we are trying to investigate and implement a double pendulum as an energy harvesting device whose chaotic movement provide energy in broadband. Solution: Let θ be the angle of the top particle with respect to the vertical, and ψthe angle of bottom particle also with respect to the vertical. The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. 2 radians and initial angular velocity dtheta/dt 0. The Euler-Lagrange Equations, a fundamental result of the calculus of variations, combined with the Principle of Least Action and the Lagrange D’Alembert Principle are used to describe the equations of motion for a cart and double-pendulum system in terms of the kinetic and potential energy of the system, which is in turn described in terms of the positions and velocities of the cart and two pendulum bobs. Numerical solution a. Let theta(t) be the corresponding angle with respect to the vertical. How Does Foucault’s Pendulum Prove the Earth Rotates? the angle of these lines changed, suggesting to audience members that the direction of the pendulum’s travel was shifting under the. Because the double pendulum problem is quite complex, Lagrangian and Hamiltonian equations are used to simplify the analysis. Damping and driving are caused by two additional forces acting on the pendulum: The damping force and the driving force. 4 (notice that θ 1 is counterclockwise, and θ 2 is clockwise!). 9) into the equation of motion of the physical pendulum (1. Simplifying assumptions can be made, which in the case of a allow the equations of motion to be solved analytically for small-angle oscillations. Hello! I'm working on my project. (I know the equations are correct) [tex](m_1+m_2)\ddot{x}+m_2l. We study the stabilization of a double pendulum mounted on a movable cart. We assume that this model will only be used when the pendulums have a relatively small angular displacement. distance between points A and B > AB := join_points(A,B): show(AB); > phi1 := dot_prod(AB,AB)-lAB^2; 2. The contact modelling is applied to a double pendulum problem. Using Lagrange equation, equation of motion of a double pendulum can be obtained and is a ordinary differential equation which is solved using Matlab ode45 solver. Key ideas: For small-angle oscillations, the motion of a simple pendulum is simple harmonic. A double pendulum released from a small initial angle behaves similarly to the single pendulum. Read and learn for free about the following article: Trig and forces: the pendulum If you're seeing this message, it means we're having trouble loading external resources on our website. Baumgarte stabilization method. 2 Double pendulum Schematic of the motion of a puck on an air table constrained by a string to. At the lowest position, speed of ball is √(20gl) Neglect any other forces on the ball except tension and gravitational force. The upper limbs are sus- pended from a mild steel rod of diameter 6 mm. Except for small-amplitude oscil-visualization of chaos in Hamiltonian systems with two degrees of freedom. (The equation of motion is a second order differential equation so its solution must have two constants of integration. Period of Oscillation of a Nonlinear Pendulum. In other words, the complex pendulum can move to anywhere within the sphere, which has the radius of the total length of the two pendulums. Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. Elastic Pend time1map. In [6]:= Copy to clipboard. Simple Pendulum is a device consisting of a small, massive body suspended by an inextensible object of negligible mass from a fixed horizontal axis about which the body and suspension are free to rotate. 10) By adding (subtracting) these equations (1. • Understand the assumptions on which the pendulum equation is based. The equations for a simple pendulum show how to find the frequency and period of the motion. The motion of a double pendulum can be periodic, quasi-periodic, or chaotic. Simulating classical motion - the double pendulum The motion of a double pendulum for a given initial energy. Figure 4 – A schematic illustrating the geometry of the chaotic double pendulum. While our derivation can be applied to. Degree of complexity and. Figure 3 – Double Pendulum. magnitude of vector u2 > phi2 := dot_prod(u2,u2)-1;. This approach is more suitable because it uses • energy, a scalar quantity, as a basis for analysis rather than force, a vector quantity, and. The ball is performing a circular motion in vertical plane. Substituting into the equation for SHM, we get. Solve the differential equation using the small-angle approximation. However, for very large energies, a regular and predictable regime is re-established (see Núñez-Yépez etal). only two independent coordinates, the angles that the first and second arm make with the vertical. How to plot double pendulum time against angle?. Summary: The equation of motion is d 2 θ dt2 + g L sinθ = 0 Go to derivation. A double pendulum released from a small initial angle behaves similarly to the single pendulum. To do this we go through the same steps. strong sensitivity to initial conditions. The equation of motion for a torsion pendulum (in small angle oscillation) is given by: Xτ : I α = −κ θ,? What is its period of oscillation 1. Contributor Jeremy Tatum (University of Victoria, Canada). For certain energies its motion is chaotic. By considering the forces acting on each of the pendulum masses, we can derive equations for the acceleration of each mass, which are our equations of motion for the double pendulum. If you're behind a web filter, please make sure that the domains *. Consider a double pendulum system. Double Pendulum. The two dots that are relevant to the tracking are the lower two. This gives me z 1= 1 z 2= 2 z 3= _ 1 z 4= _ 2: di erentiating I get z_ _ z_ _ _ = 1 _ = 2: Josh Altic Double Pendulum. Using the small angle approximation gives us. Calculated angle and corresponding x and y coordinates of the pendulum mass are exported to MATLAB workspace and plotted accordingly with timer object. Poincare sections will be created for diﬀerent initial conditions in which the physical double pendulum starts at rest. Determine the equations of motion for small angle oscillations using Lagrange's equations. One complete repetition of the motion is called a cycle. Hamiltonian systems with two or more degrees of freedom cannot always be reduced to action-angle form, giving rise to chaotic motion. motion you see is the projection of uniform circular motion onto a diameter and is called simple harmonic motion (Figure 15. While the double pendulum equations of motion can be solved relatively straightforwardly, the equations for a triple pendulum are much more involved. I'm no expert, but it was my impression that the middle pendulum gets pulled along too easily, but maybe that is just compared to the real world examples I've seen which have other variables like friction. After seeing a double pendulum harmonograph at The Museum of Natural History (in the visiting exhibit The Exploratorium) I decided to build my own. In other words, in the case of a simple pendulum, the approximation of small oscillations fully reflects the essential properties of the system. It should be relatively simple to derive the equations of motion through the Lagrangian formalism; the equations get complex very fast though. The platform movement is sudden, causing a disturbance similar to the sudden start/stop of a subway car to the standing subject. 18070sin(sqrt(4. Here we'll assume the amplitudeof the motion 0 < , so that the pendulum doesnot spin in complete circles around thepivot,but simplyoscillates back and forth. Like at 15 seconds, they seem to both start accelerating upwards. Similarly to the double pendulum example, we will use the Newton-Euler method to solve for the equations of motion. The force acting on the spring is equal to , where , is the deviation from the spring equilibrium length. Link disabled in standalone version. Chaotic Motion of a Double Pendulum. EQUATIONS OF MOTION Consider a double pendulum comprising two square plates with side length L and masses m 1 and m 2 (see Fig. A motion detector was used in the lab to study the motion of the track moving along an inclined plane plotting the velocity-time and acceleration-time graphs of the track motion. A double pendulum is a system in which a single pendulum has a second single pendulum attached at the end of the first. A double pendulum showing chaotic dynamics - the end of the pendulum is tracked by an LED. Also shown are free body diagrams for the forces on each mass. There's no need to convert them from polar coordinates. Deriving the equations of motion for the double pendulum system using method of Lagrange's Equations. We assume that. I don't know the equations of motion for a double pendulum off the top of my head, so we can derive them. point of the pendulum in presence of gravity and the case of the motion of the pendulum in weightlessness conditions under vibrations are of particular importance because no results of experiments which are related with induced stability of the double pendulum in such conditions. The angular frequency is. For small oscillations the simple pendulum has linear behavior meaning that its equation of motion can be characterized by a linear equation (no squared terms or sine or cosine terms), but for larger oscillations the it becomes very non-linear with a sine term in the equation of motion. Its equations of motion are often written using the Lagrangian formulation of mechanics and solved numerically, which is the approach taken here. To implement the above control strategy, a minor loop is closed around the little pendulum that drives the cart position to regulate the little-pendulum angleθL. 34 modeled the swing as the motion of a double pendulum. (The equation of motion is a second order differential equation so its solution must have two constants of integration. But the cool part is come from their twin nature, but the type of motion they exhibit – chaotic motion. All what you have do is wrap it in a Manipulate. (a) The Lagrangian for a spherical pendulum Let's assume that the mass is on \bottom half" of the sphere, so that the mass has a Cartesian coordinate z = lcos. Jeremiah Ruesch Department of Physics California State University, Chico Abstract: As many undergraduates are aware, the first nonlinear system often encountered is the simple plane pendulum when the small angle approximation in the equations of motion is not assumed. Double Pendulum Java Application Physics Background. We now leave the 2-body problem and consider another, rather important class of systems that can be given a complete analytic treatment. A simple pendulum consists of a single point of mass m (bob) attached to a rod (or wire) of length $$\ell$$ and of negligible weight. The motion of the pendulum can therefore be described by the polar angle , the azimuthal angle ˚, and their rates of change. This gives me z 1= 1 z 2= 2 z 3= _ 1 z 4= _ 2: di erentiating I get z_ _ z_ _ _ = 1 _ = 2: Josh Altic Double Pendulum. Summary: The equation of motion is d 2 θ dt2 + g L sinθ = 0 Go to derivation. So all you have to do is measure the (small-angle) period of oscillation of the wheel+tire, and you can compute a close approximation to the unbalanced mass. The lengths of the pendulum rods are fixed. The angle between these two velocity vectors is θ2 −θ1. Neglect friction. However, for very large energies, a regular and predictable regime is re-established (see Núñez-Yépez etal). A double pendulum consists of two small homogeneous rods, suspended at their upper ends on a horizontal shaft A. variables > q := [xB(t),yB(t),u2x(t),u2y(t)]; nq := nops(q); constraints. Constraints and Lagrange Multipliers. The equations of motion for the double pendulum are quite complex. Clockwise angles are taken positive. Pendulum L But note also that so Now this looks similar to a spring force: Identifying k with mg/L we can simply use all of our results for a mass on a spring. Find the generalized force associated with each coordinate. Determine the equations of motion for small angle oscillations using Lagrange's equations. The pendulum system used in this study transmits the rotational power of the motor that is generated as the motor rotates through the ball screw and the rotation is changed into the straight. 1, the pendulum angle is de ned from the vertical with = 0 for the straight-up, or inverted, position. For the double pendulum. If the two pendula have equal lengths and have bobs of equal mass and if both pendula are confined to move in the same plane, find Lagrange’s equations of motion for the system. Simple harmonic motion. inverted pendulum system, are implemented in form of symbolic MATLAB functions and a MATLAB GUI application. instead of one of the parameters occurring in the equations of motion. m are function les that are called. Poincare sections will be created for diﬀerent initial conditions in which the physical double pendulum starts at rest. Numerical solution a. It provides the equations that you need to calculate the period, frequency, and length of a pendulum on Earth, the. With a large. • Understand the assumptions on which the pendulum equation is based. This gives me z 1= 1 z 2= 2 z 3= _ 1 z 4= _ 2: di erentiating I get z_ _ z_ _ _ = 1 _ = 2: Josh Altic Double Pendulum. While the double pendulum equations of motion can be solved relatively straightforwardly, the equations for a triple pendulum are much more involved. One complete repetition of the motion is called a cycle. The double pendulum hangs from the center of one of the discs, and is confined to a vertical plane. If the pendulum weight or bob is pulled to a relatively small angle from the vertical and let go, it will swing back and forth at a regular period and. Double Pendulum Simulation. t t0/Ik: (10). Application of. Angle θ 2 could stay at zero but the lower mass could still be moving due to the motion of mass 1. Simplifying assumptions can be made, which in the case of a allow the equations of motion to be solved analytically for small-angle oscillations. Learn more about ode45, double pendulum, equations of motion, differential equations. The graph on the left show one angle as a function of the other evolving in time. For small oscillations the simple pendulum has linear behavior meaning that its equation of motion can be characterized by a linear equation (no squared terms or sine or cosine terms), but for larger oscillations the it becomes very non-linear with a sine term in the equation of motion. Some results of modelling are shown. For the double pendulum. doc, available for editing. Motion Chapter 14. lookat theresults. To derive the equation for simple harmonic motion, project the motion of the marker upon the diameter AB. well the double pendulum has 2 degrees of freedom, i. Find Lagrangian and the differential equation for ϴ. At this moment, we will compute dynamics equation of the double pendulum system. It should be relatively simple to derive the equations of motion through the Lagrangian formalism; the equations get complex very fast though. Using the Lagrangian to obtain Equations of Motion In Section 1. It then follows that$$ \dot\theta_i\to \epsilon\dot\theta_i\,. Its position with respect to time t can be described merely by the angle q. The uniform pulse train sent to the motor has an adjustable rate. Chaos: small Angle 1 vs. Another thought, because I'm feeling chatty: Actually finding the equation of motion of a pendulum takes differential equations. To investigate the pendulum, we need to do a controlled experiment; that is, we need to make measurements, changing only one variable at a time. By changing the length of its string, you can change its period of oscillation, T. • Know the equation of motion for a pendulum. The dynamics of the double pendulum are chaotic and complex, as illustrated below. A simple pendulum is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in Figure 1. Double pendulums are pretty awesome. Pattern Recognition. The main goal of design is to use an appropriate control method that produces suitable external torques such that the whole system can be completely stabilized and controlled. While the double pendulum is a simple physical system, you’d be hard pressed to find another device this simple that exhibits so wide a range of behavior. The pendulum is initially displaced to one side by a small angle θ 0 and released from rest with θ 0 <<1. So, the pendulum in this experiment had a period of 1 second. The idea here is that I am going to give all the details needed to determine the equation of motion (and then model it) for a basic pendulum. which is derived from the Euler-Lagrange equation, is called an equation of motion. Motion of double pendulum colliding 845 The approximate values of both angles ψ1(t∗− i), ψ2(t ∗− i) as well as the angular velocities ψ˙ 1(t∗− i),ψ˙ 2(t ∗− i) are calculated using the linear interpolation. Conversely, increasing L means the pendulum has farther to travel to reach equilibrium, reducing the angular frequency. A small change in the initial condition - and the features of the eventual trace may be quite different. Thus, s = Lθ, where θ must be measured in radians. The pendulum Consider a pendulum consisting of a mass and a massless rod. Under such circumstances long-term prognosis becomes practically impossible. Please help meAlso, the graph of the simple harmonic motion :) thank you! You want somebody to write pendulum animation code for you? And a graph of harmonic motion?. 10) By adding (subtracting) these equations (1. And the potential is. converting to a system of rst order di erential equations If I de ne new variables for 1, _ 1, 2 and _ 2 I can construct a system of four rst order di erential equations that I can then solve numerically. Double pendulum in fluid flow: 1 - double pendulum; 2 - horizontally flow Model of object The object consists of two rectangular plates 1 and 2 (Fig. Several variants of the double pendulum may be considered; the two limbs may be of equal or unequal lengths and masses, they may be simple pendulums or compound pendulums (also called complex pendulums) and the motion may be in three dimensions or restricted to the vertical plane. For simple harmonic motion to be an accurate model for a pendulum, the net force on the object at the end of the pendulum must be proportional to the displacement. If the angle is small, then the motion is. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum. for linear motion and driving in paths with small curvature. But what would be the equation of motion for ?. In our case, the inverted pendulum system is single-input, multi-output (SIMO). Elastic Pend time1map. The motion of a double pendulum can be modeled using a system of ordinary differential equations. These trajectories satisfy the equations of motion in (7) with g = 1 and E = 0. Summary: The equation of motion is d 2 θ dt2 + g L sinθ = 0 Go to derivation. This means that the force between the two pendulums is weak compared to the force of gravity on each pendulum. The mass of the cart is denoted by m. Figure 1: Double Pendulum To predict the behavior of double pendulum is very limited in certain regimes that is initial condition because the extreme sensitivity towards even small perturbations. Obtain the equations of motion of coupled pendulum using the lagrangian method. Equations of motion for an inverted double pendulum on a cart (in generalized coordinates) Consider a double pendulum which is mounted to a cart, as in the following graphic: m2 m1 l1 2 l2 q > 0 q < 01 m q y x The length of the rst rod is denoted by l1 and the length of the second rod by l2. A double-pendulum overhead crane system By using the Lagrangian method, the nonlinear dynamic model of the double-pendulum overhead crane system can be obtained as: ( + 1 + 2) ̈ +( 1 + 2) 1. Suzuki et al. Derive the equations of motion in terms of the given angles 1 and 2. General Discussion The time period for small amplitudes For very small amplitudes, the motion of the simple pendulum may be approximated by simple harmonic motion. 4, it is essentially a normal rigid pendulum with a second rigid pendulum mounted to the end of it, hence the name “double pendulum”. m are function les that are called. In classical mechanics, a double pendulum is a pendulum attached to the end of another pendulum. The full range of motion of the double pendulum is described by two complicated equations. The simplest way to keep track of "smallness" is to introduce a dummy parameter $\epsilon$ and replace $\theta\to \epsilon\theta$ everywhere. lum, double pendulum, spring pendulum, Foucault pendulum, Kater's pendulum, and spherical pendulum [1, 2]. Consider the double pendulum shown on figure 1. I'm no expert, but it was my impression that the middle pendulum gets pulled along too easily, but maybe that is just compared to the real world examples I've seen which have other variables like friction. Small oscillations of the double pendulum. Phys 7221 Hwk #9: Small Oscillations Gabriela Gonz´alez December 5, 2006 Prob 6-4: Double Pendulum We follow the conventions for angles in Figure 1. I first recognized this type of motion in a hanging bench swing but it is present to a small degree in almost everything that hangs on two strings. Modeling a Compound Double Pendulum. We will first solve for the equations of motion for the single stance and double stance phases by perform four angular momentum balances, one for each of the free body diagrams below.


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4-2i Answer by … inverse sine (arcsine) acos: inverse cosine (arccos) atan: inverse tangent (arctangent) sinh: Hyperbolic sine : cosh: Hyperbolic cosine : tanh : Hyperbolic tangent : exp: e (the Euler Constant) raised to the power of a value or expression : ln: The natural logarithm : round: round to nearest integer : floor: Returns the largest (closest to positive infinity) value that is not greater than the argument and is an integer. The calculator will generate a step by step explanation for each operation. Browser slowdown may occur during loading and creation. 1/i = ( i / i )( 1 / i ) = i / i² = i /(-1) = - i. ceil: … Inverse Complex Number. person_outlineAntonschedule 2019-07-17 19:55:41. The file is very large. Solve complex coefficient linear equation system. Download close. It is an odd function. abs: Absolute value and complex magnitude: angle: Phase angle: complex: Create complex array: conj: Complex conjugate: cplxpair: Sort complex numbers into complex conjugate pairs: i: … In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. The calculator also converts … Cartesian form of imaginary number: z = 0.2078796 Real part: x = Re z = 0.208 Imaginary part: y = Im z = 0. How to solve: How do you enter imaginary numbers on a calculator? Solvers Solvers. You can also determine the real and imaginary parts of complex numbers and compute other common values such as phase and angle. In … Complex Number Calculator. A complex number is a number that can be expressed in the form a + bi where 'a' and 'b' are real numbers and 'i' is the imaginary unit, which satisfies the equation i 2 = -1. We can evaluate the value of many complex functions by using infinite series in the same way that we can for scalar functions. Reduce the left matrix to row echelon form using elementary row operations for the whole matrix (including the right one). There are two tricks to find the multiplicative inverse of complex number. Read the instructions. To understand inverse calculation better input any example, choose "very detailed solution" option and examine the solution. Additive Inverse of Complex Numbers. Set the matrix (must be square) and append the identity matrix of the same dimension to it. Learn more Accept. The complex number calculator is also called an imaginary number calculator. Source(s): … My next calculator should be able to work with 3x3 matrices with complex numbers (determinant, invert, add, substract, multiply, ...). Favorite Answer. Gaussian elimination in complex numbers. Printable worksheet for beginning division, geometric average calculation using TI-83 plus, can you solve linear equation with only 1 known, log base 2 graph, how to find x intercept on ti calculator. Range of usual principal value. By my experience it wants to give you back an inverse Laplace with complex elements. To calculate inverse matrix you need to do the following steps. Calculation steps. Show Instructions. complex number,solving complex numbers,complex numbers problems,iota,complex numbers multiplication,complex numbers formulas,complex numbers polar form,complex,complex number,solving complex numbers,complex numbers problems,complex numbers multiplication,complex numbers calculator,complex numbers formulas,complex numbers pdf,complex numbers polar form, number … additive inverse of any number 'a' simply means a number 'b' such that a+b=0, that is b = -a. Free matrix calculator - solve matrix operations and functions step-by-step. Here you can calculate inverse matrix with complex numbers online for free with a very detailed solution. 1 0. fraleigh. By using this website, you agree to our Cookie Policy. Complex analysis. Show details. The calculator will find the inverse tangent of the given value in radians and degrees. Lv 4. Answer Save. To calculate inverse matrix you need to do the following steps. Calculate. Mixed number calculator, decimal squares, adventures in arithmetic for the pre-algebra student, year6 multiplication problems, College Algebra clep, 4th grade code math worksheets, algebra 2 answer. Set the matrix (must be square) and append the identity matrix of the same dimension to it. Algebra -> Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Find the multiplicative inverse of each complex number. You can use them to create complex numbers such as 2i+5. Solutions Graphing Practice; Geometry beta; Notebook Groups Cheat Sheets; Sign In; Join; Upgrade; Account Details Login Options Account Management Settings Subscription Logout No new … Have questions? When the result is a complex number, the calculator gives the following additional results: absolute value, argument, real part, imaginary part and a representation in the complex plane. And how does the "i" get into the numerator of the fraction? To represent a complex number, we use the algebraic notation, z = a + ib with i 2 = -1 The complex number online calculator, allows to perform many operations on complex numbers. Learn How to Modulus of complex number - Definition, Formula and Example Definition: Modulus of a complex number is the distance of the complex number from the origin in a complex plane and is equal to the square root of the sum of the squares of the real and imaginary parts of the number. As a result you will get the inverse calculated on the right. The calculator will simplify any complex expression, with steps shown. Elements must be separated by a space. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step. We just use the same series that we would use for scalar functions but plug in complex values instead of real values. Inverse tables in Electical Engineering sources provide these. Mathematics, math research, mathematical modeling, mathematical programming, math … In general, you can skip parentheses, but be very … It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. : (3-4i)*conj(3-4i). The calculator itself works with two double stacks for real and imaginary parts of numbers (stack[4] and … Rounded. This website is made of javascript on 90% and doesn't work without it. Learn more Accept. Logarithmic forms. The file is very large. By using this website, you agree to our Cookie Policy. Why does 1/i = -i? When the result is a matrix, and if it is possible, the calculator gives the following additional results: the determinant, the inverse, the trace, the scaled form and the reduced scaled form of the matrix. With this function, the calculator allows the online calculation of the imaginary part of a complex number. So if a series exists for real values we can evaluate the complex function. Complex numbers are the combination of real numbers and imaginary numbers. This calculator extracts the square root, calculate the modulus, finds inverse, finds conjugate and transform complex number to polar form. This website uses cookies to ensure you get the best experience. Why does the inverse of an imaginary number (e.g. Given a non-zero number a, its multiplicative inverse can be found out by solving for x as seen in the following equation. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Inverse hyperbolic sine value: (2.183585216564564+1.096921548830143j) Inverse hyperbolic cosine value: (2.198573027920936+1.1169261168317741j) Inverse hyperbolic tangent value: (0.09641562020299617+1.3715351039616865j) Hyperbolic sine value: (-2.370674169352002-2.8472390868488278j) Hyperbolic cosine value: (-2.4591352139173837-2.744817006792154j) … Before we go over how multiplicative inverses work for complex numbers, let's quickly review how multiplicative inverses work for real rational numbers like the ones below. Use this online algebraic conjugates calculator to calculate complex conjugate of any real and imaginary numbers. For graph, see graphing calculator. You can copy and paste the entire matrix right here. If you need to know abt multiplicative inverse, google on it, i am attaching a good link i know. As imaginary unit use, (1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8, pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225, pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303, (6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224, equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i, system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy, multiplication of three complex numbers: (1+3i)(3+4i)(−5+3i), Find the product of 3-4i and its conjugate. The domain of the inverse tangent is (-oo,oo), the range is (-pi/2,pi/2). As a result you will get the inverse calculated on the right. If a determinant of the main matrix is zero, inverse doesn't exist. 4-2i Log On Algebra: Complex Numbers Section. Operations with one complex number Five operations with a single complex number. This website uses cookies to ensure you get the best experience. Definitions as infinite series. Free tutorial and lessons. Functions. Indefinite integrals of inverse trigonometric functions. where. Now the additive inverse of A+iB should be a value, that on adding it with a given complex number, we get a result as zero. For example: Alternatively we can derive the real and imaginary parts in terms of the real functions: We can convert between the circular and hyperbolic functions as follows: 1. sin(z) = -i sinh(iz… To perform the expansion, continue as before. Complex inverse trigonometric functions. For example, the complex conjugate of X+Yi is X-Yi, where X is a real number and Y is an imaginary number. Therefore, it will be -(A+iB) … In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Each row must begin with a new line. Gaussian elimination in complex numbers Augmented matrix. Intro ... but may be easier for computer solutions (where complex numbers are handled as easily as real numbers). 4 Answers. Calculation precision. Inverse of 3x3 Matrix with Complex Numbers . The complex symbol notes i. When the result contains units, the … Exponentiation: i ^ i = 0.20787958; This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. By this way which is mentioned above it will be much easier using the calculator without involving in much math. Lv 7. You cannot use imaginary currents and voltages, if I am correct, but real currents and voltages should be obtainable. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1. 1 decade ago. A beautiful, free online scientific calculator with advanced features for evaluating percentages, fractions, exponential functions, logarithms, trigonometry, statistics, and more. Exact. Lessons Lessons. Mathematical articles, tutorial, examples. The inverse is calculated using Gauss-Jordan elimination. 4 years ago. i^3 would be i^-3) change the sign of the result? Inverse Laplace Transform, Partial Fraction Expansion. inverse sine (arcsine) of a value or expression : acos: inverse cosine (arccos) of a value or expression : atan: inverse tangent (arctangent) of a value or expression : sinh: Hyperbolic sine of a value or expression : cosh: Hyperbolic cosine of a value or expression : tanh: Hyperbolic tangent of a value or expression : exp: e (the Euler Constant) raised to the power of a value or expression : ln: The natural logarithm of a … A+iB is a complex number, where A is the real number and B is the imaginary number. Solutions Graphing Practice ; Geometry beta; Notebook Groups Cheat Sheets; Sign In; Join; Upgrade; Account Details Login Options Account Management Settings … What I need is a way around to use calculator to find the inverse for a matrix without any imaginary number inside the matrix. Because I face different size of matrices such as 3 by 3. Beginning … Answers archive Answers : Click here to see ALL problems on Complex Numbers; Question 27317: Find the multiplicative inverse of each complex number. Note that A2 and A3 must be complex conjugates of each other since they are equivalent except for the sign on the imaginary … Relevance? So, if the complex number is 'p + iq', its additive inverse is: - p - iq. Derivatives of inverse trigonometric functions. In case of complex numbers which involves a real and an imaginary number, it is referred to as complex conjugate. Inverse Imaginary Numbers? The inverse tangent y=tan^(-1)(x) or y=atan(x) or y=arctan(x) is such a function that tan(y)=x. That for I want to impement fast explicit formulae (found on Wikipedia) rather than some algorithms. Show Instructions. Hello! inverse sine (arcsine) of a value or expression : acos: inverse cosine (arccos) of a value or expression : atan: inverse tangent (arctangent) of a value or expression : sinh: Hyperbolic inverse sine (arcsine) of a value or expression : cosh: Hyperbolic inverse cosine (arccos) of a value or expression : tanh: Hyperbolic inverse tangent (arctangent) of a value or expression : exp: e (the Euler Constant) raised to the power … $$2 \text{ and } \frac 1 2$$ because $$2 \cdot \frac 1 2 = \red 1$$ Complex numbers: Calculate the multiplicative inverse for $z=a+bi \neq 0$ $a$ is the real part and $i$ is the imaginary one. To calculate the imaginary part of the following complex expression z= 1 + i 1 - i, enter imaginary_part (1 + i 1 - i) or directly (1+i)/ (1-i), if the button imaginary_part already appears, the result 1 is returned. The Inverse Laplace Transform by Partial Fraction Expansion . You need to enable it.

inverse imaginary number calculator 2021

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# Explain how to find the degree of a polynomial in two variables.

Chaya Galloway 2021-09-17 Answered
Explain how to find the degree of a polynomial in two variables.

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Usamah Prosser

Step 1
We have to explain how to find the degree of a polynomial in two variables.
Step 2
The Polynomial in two variables contains the term of the form: $$ax^ny^m$$ Now the degree of each term can be evaluated by adding the exponents of each termand the degree of the polynomial can be determined by the greatest of all those.
Consider an example:
$$\displaystyle{x}^{{2}}{y}-{6}{x}^{{3}}{y}^{{{12}}}+{10}{x}^{{2}}-{7}{y}+{3}={0}$$
First we evaluate the degree of each term Now degree of $$\displaystyle{x}^{{2}}{y}={3}$$
degree of $$\displaystyle{6}{x}^{{3}}{y}^{{{12}}}={15}$$
degree of $$\displaystyle{10}{x}^{{2}}={2}$$
degree of $$\displaystyle{7}{y}={1}$$
degree of $$\displaystyle{3}={0}$$
Now the degree of $$\displaystyle{x}^{{2}}{y}-{6}{x}^{{3}}{y}^{{{12}}}+{10}{x}^{{2}}-{7}{y}+{3}$$ is the greatest of all these So, degree of $$\displaystyle{x}^{{2}}{y}-{6}{x}^{{3}}{y}^{{{12}}}+{10}{x}^{{2}}-{7}{y}+{3}={15}$$


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# Reference frames - why do I need to convert to Geocentric Equatorial State Vector to solve equation of motion?

I am asking this question in relation to an example shown in the 'Orbital Mechanics for Engineering Students' book by Howard Curtis (third edition) pp659 example 12.1.

The example is demonstrating how to calculate the orbit decay given the Keplerian elements and ballistic coefficient. The process goes something like this:

Convert Keplerians Elements ==> [R, V] State Vector in perifocal frame ==> [R, V] state vector in Geocentric equatorial frame ==> numerical solution of equation of motion:

$$\ddot{\mathbf{r}} = -\mu \frac{\mathbf{r}}{r^3} + p$$

where p is the perturbing drag acceleration:

$$\mathbf{p} = -\frac{1}{2} \rho \upsilon^2 \left(\frac{C_D A}{m} \right) \mathbf{v_{rel}}$$

After numerical solution of the ODE, a plot of altitude vs. time is generated.

My question is: why is it necessary to convert the Keplerian elements into a state vector in the geocentric equitorial frame. I get that a state vector representation is needed to work with the equations of motions, but why can't you use just solve the equations of motions with the state vectors in perifocal frame? Is there a particular reason/advantage of doing this?

• It looks like this is an exercise meant to be educational. It doesn't really say it is necessary, does it?
– uhoh
Mar 17 '18 at 13:00
• Also, I've changed your equations to MathJax which is the supported way to render equations here (rather than the external site you linked to). Are you sure the second equation is correct? Shouldn't it be just $\upsilon \mathbf{v}$ instead of $\upsilon^2 \mathbf{v}$? Check your units.
– uhoh
Mar 17 '18 at 13:15

I'm not a specialist, but here are my guesses: the geocentric equatorial frame makes it easier to express $\mathbf{v}_{rel}$, the spacecraft's velocity relative to the atmosphere, in terms of $\mathbf{r}$ and $\mathbf{v}$: $$\mathbf{v}_{rel} = \mathbf{v} - \boldsymbol{\omega}_E\times \mathbf{r},$$ where $\boldsymbol{\omega}_E$ is the angular velocity of Earth's rotation, which points in the $z$-direction.
Well, if by perifocal frame you mean the one for the initial orbit, with the directions of the axes staying the same (I'm not sure whether that's how this term is used), then the formula for $\mathbf{v}_{rel}$ is the same. The only difference is that $\boldsymbol{\omega}_E$ doesn't point in the direction of an axis anymore, you need to calculate its direction first. But sure, you can solve the equations in this frame as well. Note, however, that since $\mathbf{v}_{rel}$ and the drag don't have to lie in the orbit's plane, the plane may change over time, which decreases the usefulness of the perifocal frame (it won't be perifocal for later orbits).


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# A little confused

#### vanillasnow

book states given ln(3x+1) h(x)e^-3x G(x)
3^x

find d/DC fx+he+gx

then find d/Dx x^2 (f(x)

how how do I solve this?

doni get the derivative to all and then add to the first problem?

for second: do I get the derivative of f(X) and multiply to x^2 OR do I take derivative to x^2 before multiplying?

#### Debsta

MHF Helper
I can't make sense of your question to help you. Could you make it clearer. Use brackets if necessary.

#### skeeter

MHF Helper
book states given ln(3x+1) h(x)e^-3x G(x)
3^x

find d/DC fx+he+gx

then find d/Dx x^2 (f(x)
a little confused? ... so am I

Did you forget some equal signs?

Is $f(x)=\ln(3x+1)$ ?

Is $h(x)=e^{-3x}$ ?

Is $g(x)=3^x$

What does d/DC mean? Do you really mean $\dfrac{d}{dx} \bigg[f(x)+h(x)+g(x)\bigg]$ ?

If so, then the derivative of a sum of functions is the sum of their derivatives ... $f'(x)+h'(x)+g'(x)$

Is the second problem $\dfrac{d}{dx} \bigg[x^2 \cdot f(x) \bigg]$ ?

If so, use the product rule ... I'm sure you were taught that since you've been asked to do it.

In future, please proofread your post by using the "preview" button, make any corrections, preview again, then submit. You'll find folks here are more inclined to help if they can read and understand what you posted. Avoid using phone text abbreviations. Thank you for your cooperation.

#### vanillasnow

First of all,

Thank you for the help. Secondly, my computer would not work and as you mentioned, was done one the phone. Unfortunately I was not able to make corrections yesterday and with the limited view screen, It was pretty hard to proof read as everything I typed had a time delay.

But yes, you retyped correct, and it was what I was asking.

Now to your question: product rule for second problem.
You'd think we ( the entire) class would know, but unfortunately we are faced with a math teacher who gives very little explanation.

In other words, when one looks at the problem at hand,problem 2- in where you pointed to use the product rule, do we first take the derivative of f'(x) and plug it (multiply it by (x^2) OR take the derivative of (x^2) as well and use the product rule...?

Thank you again, Jen

#### skeeter

MHF Helper
Product rule ...

$\dfrac{d}{dx} \bigg[u(x) \cdot v(x)\bigg] = u(x) \cdot v'(x) + u'(x) \cdot v(x)$

$\dfrac{d}{dx} \bigg[x^2 \cdot f(x)\bigg] = x^2 \cdot f'(x) + 2x \cdot f(x)$

#### vanillasnow

Thank you !!! you are excellent


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# Graphical analysis and Visualization for LPP

$$Q4.$$Use graphical analysis to find the optimal solution(s) of the LPP

a)

Maximize $$3x+2y$$

subject to $$3x-4y\le11$$ $$x,y\ge0$$

$$b)$$

Maximize $$5x-7y$$

subject to $$x+y\le18$$ $$x+3y\ge-6$$ $$x,y\ge0$$

$$Q5.$$ Visualizing canonical form problems

a) In $$\mathbb{R}^3$$ draw the plane $$P:x_1+x_2+x_3=1$$ inside $$P$$ draw the set where $$x_1,x_2,x_3\ge0.$$ Also inside $$P$$ draw the line where $$x_1=0,x_2=0,$$ and $$x_3=0.$$

b) In $$\mathbb{R}^3$$ draw the plane $$P:3x-4y+s=11$$.Inside $$P$$ draw the set where $$x,y,s\ge0$$ and the line $$x=0,y=0,$$ and $$s=0$$.

$$Q4.$$

a)

The darkest region contains all the feasible solution, and the blue line is the objective function, as $$z=3x+2y$$ increasing, the blue line keeps moving up, but some parts of the line will always in the feasible region, therefore it doesn't have max feasible solution in this case.

b)

The darkest region contains all the feasible solution, and the blue line is the objective function, as $$z=5x-7y$$ increasing, the blue line keeps moving down, that the blue point is the max feasible solution in this case.$$x=18,y=0$$

$$Q5.$$

$$a)$$

This is how $$P$$ looks like, The graph didn't show the parts that go though the $$x_1,x_2$$ plane, , but the plane does go though, basicly $$P$$ is a plane that has intersections $$(1,0,0),(0,1,0)$$ and $$(0,0,1)$$ on $$x,y,z$$-axis

And the blue triangle is the parts greater then zero, in another words $$x_1,x_2,x_3\ge0$$ in $$P$$

The red line is union of $$x_1,x_2,x_3=0$$ in $$P$$

$$b)$$

This is $$P$$, The graph didn't show the parts that go though the $$x,y$$ plane, but the plane does go though.

This is $$x,y,s\ge0$$ in $$P$$, which is not bounded.

The red line is union of $$x,y,s=0$$ in $$P$$


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# MATLAB: Power spectral density of a signal

power spectral density

Hello I have EEG dataset in excel format with time and voltage values.I need to plot the power spectral density of the signal. I have loaded the excel file in Matlab and plotted the voltage vs time values. Now I need to calculate the power spectral density. Generally the frequency range of EEG signals between 0-30 Hz. Since I have only voltage and Time values, I do not know the frequency of my signal. I need to calculate Power spectral density Using FFT. I am including the code in matlab file and time voltage values of the signal in txt file for reference. Please suggest me.
dataset=xlsread('input signal.xlsx','sheet1','A1:B770');t = dataset(:,1);V = dataset(:,2);X=[t V];subplot(2,2,1);plot(X(:,1), X(:,2))xlabel ('time');ylabel('voltage');title('original Signal');

h1=spectrum.welch;set(h1,'Windowname','Hann');Fs=1000;set(h1,'OverlapPercent',66.7);set(h1,'SegmentLength',512);myPsd=psd(h1,X(:,2)-mean(X(:,2)),'Fs',Fs)semilogx(myPsd.Frequencies,myPsd.Data);grid on


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3 Tutor System
Starting just at 265/hour

# Calculate the wavelength of a sound wave whose frequency is 220 Hz and speed is 440 m/s in a given medium.

Given that,

Frequency of sound wave = 220 Hz.

Speed of sound wave = 440 m/s.

Calculate wavelength.

We know that,

Speed = Wavelength × Frequency

$$\Rightarrow v = \lambda \nu$$

$$\Rightarrow 440 \ = \ Wavelength \ × \ 220$$

$$\Rightarrow Wavelength \ = \ \frac{440}{220}$$

$$\therefore$$ Wavelength = 2 m


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# How to solve this kind of Conditional Probability problems?

I have a series of problems that involve events that happen one after another, and where the usual formulas like $P(\frac{A}{B})=\frac{P(AB)}{P(A)}$ don't seem to help.

Example:

In a horse race there are $3$ horses. $E12$ means horse $1$ finishes before horse $2$. $E123$ means horse $1$ finishes before $2$, which finishes before $3,$ etc.

If $P(E13)=2/3$ and $P(E23)=1/2$ then $P(E123)=?$

How can I go about solving this kind of problems?

-
@GeorgeS Please refrain from petty unnecessary edits on 4+ years old posts – Did May 11 '15 at 15:57
Sure. I was checking out older questions and it was bothering me. – George Simpson May 11 '15 at 15:58

This is not a conditional probability question. Your particular example has insufficient information to solve it. Here $E123$ is the event of the horses finishing in the order 1, 2, 3. So (assuming the horses can't tie for a position) $$P(E13)=P(E132)+P(E123)+P(E213),$$ $$P(E23)=P(E123)+P(E213)+P(E231)$$ and of course $$1=P(E123)+P(E132)+P(E213)+P(E231)+P(E312)+P(E321).$$ Knowing $P(E13)$ and $P(E23)$ isn't enough to determine any one of the $P(Eijk)$s.


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# Wrapping up GSoC 2019

Published:

Summary: The final blog for GSoC 2019.

# Introduction

With GSoC 2019 coming to an end, this is my final blog which mentions all my work for the project Rubyplot.

# What is Rubyplot?

RubyPlot is a plotting library in Ruby for scientific development inspired by the library Matplotlib for Python. Users can create various types of plots like scatter plot, bar plot, etc. and can also create subplots which combine various of these plots. The long-term goal of the library is to build an efficient, scalable and user-friendly library with a backend-agnostic frontend to support various backends so that the library can be used on any device.

## Examples

Creating graphs in Rubyplot is very simple and can be done in just a few lines of code, for example:

require 'rubyplot'

figure = Rubyplot::Figure.new(width: 30, height: 30)

axes00.plot! do |p|
d = (0..360).step(5).to_a
p.data d, d.map { |a| Math.sin(a * Math::PI / 180) }
p.fmt = 'ok-'
p.marker_fill_color = :white
p.marker_size = 0.5
p.line_width = 2
p.label = "sine"
end

axes00.title = "A plot function example"
axes00.square_axes = false

figure.write('example1.png')

Has the output:

require 'rubyplot'

figure = Rubyplot::Figure.new(width: 30, height: 30)

axes00.candle_stick! do |p|
p.lows = [0, 10, 20, 30, 20, 10]
p.highs = [40, 50, 60, 70, 60, 50]
p.opens = [10, 20, 30, 40, 30, 20]
p.closes = [30, 40, 50, 60, 50, 40]
p.color = :yellow
end
axes00.candle_stick! do |p|
p.lows = [5, 5, 25, 30, 10, 10]
p.highs = [50, 40, 65, 70, 80, 60]
p.opens = [10, 20, 30, 40, 30, 20]
p.closes = [35, 35, 45, 60, 75, 50]
p.color = :blue
end

axes00.title = "A multi candle-stick plot"
axes00.square_axes = false

figure.write('example2.png')

Has the output:

require 'rubyplot'

figure = Rubyplot::Figure.new(width: 30, height: 30)

axes00.bar! do |p|
p.data [1, 2, 3, 4, 5]
p.color = :lemon
p.spacing_ratio = 0.2
end
axes00.bar! do |p|
p.data [5, 4, 3, 2, 1]
p.color = :blue
end
axes00.bar! do |p|
p.data [3, 5, 7, 5, 3]
p.color = :red
end

axes00.title = "A multi bar plot"
axes00.square_axes = false

figure.write('example3.png')

Has the output:

# History of Rubyplot

Rubyplot started as two GSoC 2018 projects by Pranav Garg(@pgtgrly) and Arafat Dad Khan(@Arafatk) and the mentors from The Ruby Science Foundation(SciRuby), Sameer Deshmukh(@v0dro), John Woods(@mohawkjohn) and Pjotr Prins(@pjotrp). Pranav Garg worked on the GRRuby which had the GR backend and Arafat Dad Khan worked on Ruby Matplotlib which had the ImageMagick backend. The ultimate goal of combining both and creating Rubyplot. After GSoC 2018, Sameer Deshmukh combined both projects and created Rubyplot and he has maintained it ever since. Around May 2019, I started working on Rubyplot as a part of GSoC 2019.

# GSoC 2019

As a part of GSoC 2019, my project had 3 major deliverables:

1. ImageMagick support(Phase 1): Support for ImageMagick back-end will be added in addition to the currently supported back-end GR, the front-end of the library will be back-end agnostic and the current overall integrity of the library will be preserved.
2. Plotting and show function(Phase 2): A new plot function will be added which plots markers (for example circles) to form a scatter plot with the points as inputs (same as plot function in Matplotlib). A new function show will be added which will allow viewing of a plot without saving it. This plot function will be back-end agnostic and hence will support both GR and Magick back-end.
3. Integration with iruby notebooks(Phase 3): Rubyplot will be integrated with iruby notebooks supporting all backends and allowing inline plotting.

As a part of GSoC 2019, I completed all the deliverables I had initially planned along with a tutorial for the library and some other general improvements.
Details of my work are as follows:

## Phase 1

During Phase 1, I focused on setting up the ImageMagick backend which involved the basic functionality required for any backend of the library which are X-axis and Y-axis transform functions, within_window function which is responsible for placing the plots in the correct position, function for drawing the X and Y axis, functions for drawing the text and scaling the figure according to the dimensions given by the user. I implemented these functions using internal rmagick functions which were very useful like scale, translate, rotate, etc.
After this, I worked on the scatter plot, which was the first plot I ever worked on. This plot had a very particular and interesting problem, which was that different types of markers were internally implemented in the GR backend, but for ImageMagick backend, I had to implement everything using basic shapes like circles, lines, polygons and rectangles. To solve this I created a hash of lambdas which had the code to create different types of markers using the basic shapes.
After this I implemented all the simple plots which Rubyplot supports, these are line plot, area plot, bar plot, histogram, box plot, bubble plot, candle-stick plot and error-bar plot.

So, during Phase 1, I completed the following deliverables -

1. Set up the ImageMagick backend to have the basic functionality.
2. Implemented and tested the simple plots in Rubyplot which are scatter plot, line plot, area plot, bar plot, histogram, box plot, bubble plot, candle-stick plot and error-bar plot.

Code for Phase 1 can be found here.

## Phase 2

I started Phase 2 by implementing the multi plots which are multi stack-bar plot, multi-bar plot, multi-box plot and multi candle-stick plot.
Next, I implemented the plot function which is a combination of scatter plot and line plot, using the plot function the user can easily create a scatter plot or a line plot or a combination of both. The most interesting feature of the plot function is the fmt argument which sets the marker type, line type and the colour of the plot using just characters, so instead of writing the name of the type and setting the variables, the user can simply input a string in fmt argument which has the characters for corresponding marker type, line type and colour.
Next was to implement the show function which is an alternative to write function. It draws the Figure and shows it on a temporary pop-up window without the need of saving the Figure on the device, this allows the user to test the code quickly and easily. This was done by using internal functions of the backends which are display for ImageMagick and gr_updatews for GR.

So, during Phase 2, I completed the following deliverables -

1. Implemented and tested the multi plots in Rubyplot which are multi stack-bar plot, multi-bar plot, multi-box plot and multi candle-stick plot.
2. Implemented and tested the plot function with fmt argument.
3. Implemented and tested the show function.

Code for Phase 2 can be found here and here.

## Phase 3

During Phase 3, I integrated Rubyplot with the IRuby notebooks which allow the user to draw figures inside the notebook just by using the show function, through this integration the user can quickly and easily test the code step by step before running the whole codebase.
I also implemented ticks for ImageMagick backend.
Finally, I created a tutorial for the library which also contains template codes for all the plots which a user can easily get familiar with the working of the library and start using it.

So, during Phase 3, I completed the following deliverables -

1. Integrated Rubyplot with IRuby notebooks with the support for inline plotting.
2. Implemented and tested ticks for Magick backend.
3. Created the tutorial for Rubyplot.

Code for Phase 3 can be found here.

# Resources(blogs, code, etc.)

## Previous Work

• GSoC 2018 project GRRuby by Pranav Garg can be found here
• GSoC 2018 project Ruby Matplotlib by Arafat Dad Khan can be found here
• A talk on Rubyplot by Pranav Garg in RubyConf 2018 can be found here

## My work

• Daily updates can be found here
• Proposal can be found here
• Tutorial notebook can be found here and can be viewed online(rendered) here
• Rubyplot Github Repository can be found here
• All my work can be found in these PRs: PR#45 and PR#52
• Other blogs can be found here:

# Future Work

I plan to keep contributing to Rubyplot and also start contributing to other projects of SciRuby.
Future work to be done for Rubyplot is to write documentation, add more tests, add more types of plots, add more backends, make the plots interactive and in future add the feature for plotting 3-Dimensional graphs which would also be interactive.

# EndNote

With this, we come to an end of GSoC 2019. These 3 months have been very challenging, interesting, exciting and fun. I got to learn a lot of things while working on Rubyplot and while interacting with my mentors. I have experienced an improvement in my Software development skills and programming in general which will help me a lot in future. I would love to keep working with SciRuby on more such interesting projects and maybe even try for GSoC again next year ;)

# Acknowledgements

I would like to express my gratitude to my mentor Sameer Deshmukh for his guidance and support. He was always available and had solutions to every problem I faced, I got to learn a lot from him and I hope to learn a lot more from him in the future. I could not have asked for a better mentor.

I would also like to thank Pranav Garg who introduced me to Ruby and also to the SciRuby community. During his GSoC 2018 project, he introduced me to the Rubyplot library and helped me get started with it. His suggestions were very helpful during my GSoC 2019 project.

I would also like to thank mentors from SciRuby Prasun Anand and Shekhar Prasad Rajak for mentoring me and organising the occasional meetings and code reviews. I would also like to thank Udit Gulati for his helpful insights during the code reviews.

I am grateful to Google and the Ruby Science Foundation for this golden opportunity.

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# Why Functional Programming Matters

Why Functional Programming Matters John Hughes, Research Topics in Functional Programming, 1990 (based on an earlier Computer Journal paper that appeared in 1989).

1989/1990 must have been a fairly dark time to be a functional programmer. Object-oriented programming was rising in prominence and the dream that industry would pay attention to functional programming looked like it was getting further away, not closer. What a time then to publish “Why Functional Programming Matters,” John Hughes’ powerful argument for why the mainstream might be making a mistake in ignoring functional programming. You can almost hear the lone voice in the corner saying “hey, over here!”

This paper is an attempt to demonstrate to the larger community (of non-functional) programmers the significance of functional programming, and also to help functional programmers exploit its advantages to the full by making it clear what those advantages are.

The version of the paper I’m referencing runs to 23 pages, but the bulk of that is examples illustrating the author’s points. The central argument itself is fairly concise, and most gloriously argued, and tying in with a theme we’ve been bouncing around over the last week or so, it all comes back to modularity.  You’ll have to forgive me for extensive quoting from the opening sections of the paper today, Hughes’ prose is just too good not to!

It is now generally accepted that modular design is the key to successful programming… However, there is a very important point that is often missed. When writing a modular program to solve a problem, one first divides the problem into subproblems, then solves the subproblems, and finally combines the solutions. The ways in which one can divide up the original problem depend directly on the ways in which one can glue solutions together. Therefore, to increase one’s ability to modularize a problem conceptually, one must provide new kinds of glue in the programming language.

But we’re getting ahead of ourselves… When most people sum up the advantages of functional programming argues Hughes, they talk about the fact that functional programs are side-effect free and contain no assignment statements. Expressions can therefore be evaluated at any time and replaced by their values, and vice-versa – programs are “referentially transparent.” People still talk a lot about the value of immutability and side-effect free computing today, and of course these things have value. Yet it’s not a good way to explain FP to non-believers says Hughes:

Such a catalogue of “advantages” is all very well, but one must not be surprised if outsiders don’t take it too seriously. It says a lot about what functional programming isn’t (it has no assignment, no side effects, no flow of control) but not much about what it is. The functional programmer sounds rather like a medieval monk, denying himself the pleasures of life in the hope that it will make him virtuous. To those more interested in material benefits, these “advantages” are totally unconvincing.

Oh, but there are benefits says the functional programmer, functional programs are an order of magnitude shorter and therefore functional programmers are more productive.

Yet why should this be? The only faintly plausible reason one can suggest on the basis of these “advantages” is that conventional programs consist of 90% assignment statements, and in functional programs these can be omitted” This is plainly ridiculous. If omitting assignment statements brought such enormous benefits then FORTRAN programmers would have been doing it for twenty years.

If this characterization of functional programming is inadequate, what is it that can both explain the power of functional programming and also give a clear indication of what the functional programmer should strive towards. Think back to the time when structured programming was new (you remember that right?) the advantages says Hughes were summed up more or less as “structured programs contain no goto statements.” This is similar to the negative ‘advantages’ advocated for functional programming.

With the benefit of hindsight, it’s clear that these properties of structural programs, although helpful, do not get to the heart of the matter. The most important difference between structured and unstructured programs is that structured programs are designed in a modular way. Modular design brings with it great productivity improvements:

• Small modules can be coded quickly and easily
• General purpose modules can be reused, leading to faster development of subsequent programs
• The modules of a program can be tested independently, helping to reduce the time spent debugging.

The absence of gotos helps with programming ‘in the small’, but modular design helps with programming ‘in the large.’ And now we’re back at the opening quote: “The ways in which one can divide up the original problem depend directly on the ways in which one can glue solutions together.”

We shall argue in the remainder of this paper that functional languages provide two new, very important kinds of glue… This is the key to functional programming’s power – it allows improved modularization. It is also the goal for which functional programmers must strive – smaller and simpler and more general modules, glued together with the new glues we shall describe.

The two new glues are

• Higher-order functions, and
• Lazy evaluation

Higher-order functions enable simple functions to be glued together to make more complex ones. I’m going to assume most readers are familiar with this idea. The example given in the paper is foldr, which captures a common computation pattern over lists, such that we can write for example:

sum = foldr (+) 0
product = foldr (*) 1
anytrue = foldr or False
alltrue = foldr and True
length = foldr count 0 // where count a n = n + 1
map f = foldr (Cons .f) Nil
summatrix = sum . map sum

And many more examples (the programming language used in the paper is Miranda).

These examples should be enough to convince the reader that a little modularization can go a long way. By modularizing a simple function (sum) as a combination of a higher-order function and some simple arguments, we have arrived at a part (foldr) that can be used to write many other functions on lists with no more programming effort.

This doesn’t just apply to lists of course, you can build higher order functions for working with all kinds of data structures.

All this can be achieved because functional languages allow functions that are indivisible in conventional programming languages to be expressed as a combinations of parts — a general higher-order function and some particular specializing functions. Once defined, such higher-order functions allow many operations to be programmed very easily. Whenever a new datatype is defined, higher-order functions should be written for processing it. This makes manipulating the datatype easy, and it also localizes knowledge about the details of its representation.

Lazy evaluation requires a little more thinking to see why Hughes classes it as a modularization mechanism: lazy evaluation makes it practical to modularize a program as a generator that constructs a large number of possible answers, and a selector that chooses the appropriate one. Without lazy evaluation this would not always be practical (or even possible, in the case of infinite generators).

We have described lazy evaluation in the context of functional languages, but surely so useful a feature should be added to non-functional languages – or should it? Can lazy evaluation and side-effects coexist? Unfortunately they cannot: adding lazy evaluation to an imperative notation is not actually impossible, but the combination would make the programmer’s life harder rather than easier.

Something to think about for those languages that like to blend functional and non-functional constructs…

Why exactly does it make the programmer’s life harder though?

Because lazy evaluation’s power depends on the programmer giving up any direct control over the order in which the parts of a program are executed, it would make programming with side effects rather difficult, because predicting in what order – or even whether – they might take place would require knowing a lot about the context in which they are embedded. Such global interdependence would defeat the very modularity that – in functional languages – lazy evaluation is designed to enhance.

There follows in the paper a series of examples demonstrating the power of lazy evaluation and higher-order functions: Newton-Raphson square roots; numerical differentiation and integration; and an alpha-beta heuristic for evaluating game trees. In all cases it’s impressive how quickly very powerful and expressive levels of abstraction are reached, and they are well worth studying.


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## Introduction

The digestive tract of vertebrates houses an extraordinary number and variety of microorganisms including bacteria, viruses, fungi, and archaea which have co-evolved with their host, and are collectively referred to as the gut microbiota. Over the past two decades, researchers have tried to understand its implications for the host’s health, and its primary role in maintaining gut homeostasis. Commensal bacteria naturally present in the human intestinal tract are now well recognized to confer several beneficial functions to the host, not only for the digestion of food products but also for protecting from pathogenic microbial infections, and shaping the mucosal immune system1,2.

Most of the studies focusing on the gut microbiota of vertebrates have been conducted in humans, for which the medical challenge is of tremendous relevance3. However, little is known about the intestine microbiota of marine vertebrates, such as fish or mammals, despite their crucial ecological role and economical/societal values. The few information available suggests that the microbial network in fish gastrointestinal tract is very complex and probably plays a vital role in fish nutrition and disease prevention4,5. However, we still lack of clear information on how the microbial equilibrium is regulated within the intestine. Indeed, vertebrates need a stable, balanced gut bacteriome to be healthy6, and recently there has been increasing interest in understanding the factors that shape the activity and composition of enteric bacteria, as well as their subsequent role in host physiology and immunity.

Researchers discovered that viruses infecting bacteria (ie, phages) are highly abundant and diversified within the human gut, and forming key components of this specific microflora7,8,9. However to our knowledge, the occurrence, composition, and life traits of such enteric viruses have never been rarely explored in marine organisms for which they could also could play pivotal role, through lytic and/or lysogenic control of their bacterial hosts. For instance there is no information about viral abundances in the digestive tract of fishes, their infection strategies, nor their susceptibilty to ambient environmental conditions (salinity, temperature, xenobiotics, etc.). The only information available refers to the diversity of culturable phages in freshwater fishes (He & Yang 2015), which only represent a small fraction of the whole gastrointestinal community of viruses10.

In the present study, we examined the main ecological features of viruses present in the gut of the black-chinned tilapia (Sarotherodon melanotheron). This fish species was selected as a potential sentinel because of its abundance, physiological properties, and wide distribution in tropical coastal ecosystems. Indeed, this extreme euryhaline species, which is endemic to West Africa, is able to live in all types of water bodies, from freshwater to marine and even hypersaline waters11.

Individuals were collected from various sites in Senegal covering a salinity range from 3 to 104‰, and exhibiting contrasted concentrations of xenobiotics. This sampling strategy allowed us to (i) acquire basic information about the abundance, life strategies and morphological composition of viral communities found in the fish gut, and (ii) explore whether such viral communities are affected by salinity and/or the presence of persistent pollutants in their environment.

## Material and Methods

### Sampling sites

The study was conducted in Senegal (West Africa) during the dry season (May 2010) at 8 contrasted stations, namely Niayes 1 (NE1), Niayes 2 (NE2), Koular (KUE), Kaoloack (KOE), Koïlal (KOI), Hann Bay (BHE), Foundiougne (FE), and Missirah (ME) (Fig. 1). These sites are located in the Dakar peninsula and along the estuary of Sine-Saloum, covering a salinity gradient from 3 to 104‰ (see Table 1).

At each station, water salinity was measured with a refractometer. Surface sediment samples were also collected using a sediment corer (core tube 70 mm inner diameter; maximum sediment penetration depth of 6 cm), which allows collecting samples hermetically sealed. Visual inspection of the overlying waters and sediment surfaces revealed very limited resuspension effects during sampling. Sediment samples were then stored in aluminum containers and frozen at −20 °C for further chemical analysis of persistent organic pollutants such as polychlorinated biphenyls (PCB), organochlorine pesticides (OCP), and polycyclic aromatic hydrocarbons (PAH)12.

### Extraction of fish gut

At each sampling site, three adult tilapia Sarotherodon melanotheron heudelotii of similar size (130–150 mm fork length) were caught using a cast net and were anaesthetized progressively in icy water. Fish were then killed by cervical dislocation (following the European directive 2010/63/UE), as there is no regulation or guidelines in Senegal where experiments were carried out. Captured fish were individually placed in plastic bags and immediately stored in ice before dissection (within 5-h after sampling)13. Briefly, the digestive tract was extracted from each individual and cut from below the stomach to the rectum. Each gut was squeezed to expel content, which represented a minimum volume of 3 ml.

At each stations, triplicate raw gut contents (RGC) were processed for the different analyses as follows: (i) 1 ml was immediately processed for estimating the proportion of bacterial cells under lysogenic infections, and (ii) 2–3 ml were fixed with formaldehyde (final concentration f.c. 3% v/v), flash-frozen in liquid nitrogen and stored at −80 °C until measurement of viral and bacterial abundances, and distribution of the viral morphotypes. Triplicate subsurface water samples (20 ml) were also collected at each station, fixed with formaldehyde (f.c. 2% v/v) and stored at −80 °C in liquid nitrogen for determination of the parameters described above.

### Enumeration of viruses and prokaryotes

For each replicate, 1 mL of the fixed RGC was suspended in 1 mL of a 9‰ NaCl sterile saline solution complemented with 0.2 mL pyrophosphate (0.1 M) to extract bacterial and viral particles from gut matrix13. After vortexing for 2 min at room temperature, the supernatant was immediately transferred into a sterile tube containing 10 mL of 9‰ NaCl. The number of viruses and prokaryotes contained in each replicate samples was determined after retention of the particles on 0.02 µm pore-size membranes (Anodisc) and staining with SYBR Gold14. On each slide, 300–500 bacteria and viruses per replicate were counted with an Olympus Provis-AX70 epifluorescence microscope, under blue light excitation (488 nm). For planktonic viruses and bacteria, since the extraction procedure was not necessary, we only applied the staining protocol, as described above, by using a volume of 300–500 µL of seawater.

### Fraction of lysogenic bacteria

We used the method of Jiang and Paul15 to initiate prophage induction in gut and planktonic bacteria. Mitomycin C (final concentration: 1 mg mL−1, Sigma-Aldrich) was added to triplicate 1-mL volumes of unfixed and fresh RGC and water. Triplicate untreated samples served as control. All sub-samples were formalin fixed (f.c., 3% [vol/vol]) after being incubated for 12 h in the dark, at in situ temperatures (eg between 25.3–29.3 °C). Prophage induction was calculated as the difference in viral abundance (epifluorescence counts, see above) between mitomycin C-treated (Vm) and control incubations (Vc). The fraction of lysogenic bacterial cells (FLC) was calculated as follows:

$${\rm{F}}{\rm{L}}{\rm{C}}\,({\rm{ \% }})=100\times ({\rm{V}}{\rm{m}}-{\rm{V}}{\rm{c}})/{\rm{B}}{\rm{S}}\times {{\rm{B}}{\rm{A}}}_{{\rm{t}}0},$$

where BS is the burst size (number of viruses per bacterial cell) and BAt0 is the bacterial abundance at the start of the experiment, i.e., before adding mitomycin C16. Since no burst size from infected gut bacteria has been published so far, we used a burst size of 24, which is an average value calculated from the range for different studied marine environments17.

### Examination of viral morphotypes

The distribution of the different viral morphotypes in gut and water samples was examined using transmission electron microscopy (TEM)18. Viruses from 500 µL aliquots of formalin fixed and pyrophospate-treated samples (see above) were harvested by repeated ultracentrifugation of 50 µL onto grids (400 mesh Cu electron microscope grids with carbon coated Formvar film) using an A-100/30 rotor in an air-driven ultracentrifuge (Airfuge®, Beckman) at 105,000 × g for 70 min. The grids were then stained for 30 s with uranyl acetate (2%, w/w), and viruses were examined and measured using a JEOL 1200EX TEM operated at 80 kV and magnification from x20,000 to x100,000. Between 300 and 400 viruses were examined per grid. Three morphotypes were distinguished for shape classification of tailed viruses (Caudovirales). Tailed viruses with isometric heads and long non-contractile tails were considered to belong to the family Siphoviridae. Tailed viruses with isometric heads and contractile tails (presence of a neck) to the family Myoviridae. Tailed viruses with isometric heads and short tails to the Podoviridae. The proportion of tailless icosahedral viruses was also evaluated.

### Data analysis

Data were log transformed to satisfy the requirements of normality and homogeneity of variance necessary for parametric analyses. Simple relationships between original data sets were tested using Pearson correlation analysis. Statistical comparisons of mean values were performed using one-way Anova and Tukey post hoc tests. All statistical analyses were performed using SIGMASTAT software.

## Results and Discussion

The digestive tract of S. melanotheron harbored a highly concentrated community of viruses and bacteria, ranging respectively from 0.2 to 10.7 × 109 viruses ml−1, and from 0.1 to 6.8 × 109 bacterial cells ml−1 of gut content (Table 2). This is to our knowledge, the first report of viral abundances in the fish gut. Such abundances fall within the same order of magnitude of those commonly found in human gastrointestinal tracts (e.g., around 109 viruses g−1)8,19, but are much greater than those reported in natural environments such as marine or even soil ecosystems, which rarely surpass 109 viruses per milliliter of water or gram of soil20,21. Whilst the main information available on gut-associated viruses refers to those examined in humans9,22,23, the positive and highly significant correlation between viral and bacterial concentrations suggested that the fish gut viral communities were dominated by phages (Pearson correlation r = 0.90, p < 0.05, Table 3). This was also corroborated by the important proportion of tailed viruses of the Caudovirales families (Sipho-, Myo- and Podoviridae), which was significantly higher in the fish gut than in the surrounding water (Table 3). Given that the Firmicutes, Proteobacteria and Bacteroidetes typically represent up to 90% of the fish intestinal microbiota4, one may suspect that these abundant tailed and temperate phages are specific of these particular bacterial groups. Interestingly, similar bacteria prevalence were reported in the human gastrointestinal tract1,24 strongly supporting the paradigm of a core gut microbiome comprised of common bacterial members ensuring the primary functions of this vital organ (e.g., defense against pathogen, etc.). By analogy, one may also suspect the existence of a core viral diversity in vertebrate gut. The rest of the microbial community being rather dedicated to the specific digestion processes and diet, which are highly species-specific.

The mitomycin-C inductions revealed a high proportion of lysogenic cells in the fish gut (min-max, 8.1–33.0%), which was significantly higher than that was found in their planktonic counterparts (min-max, 0.3–8.8%) (Table 2). However, incubations were performed ex vivo and therefore the results should be taken with caution. At first sight, such results are surprising as lysogeny is usually considered as a refuge strategy for viruses in habitats where the conditions for their reproduction is threatened by either low abundance and/or compromised host metabolism25,26. Yet, the gut of S. melanotheron contained a dense population of bacteria (eg, 0.1–6.8 × 109 cells ml−1), resulting in a rather low virus-to-bacteria ratio (VBR) (mgut = 1.5 ± 0.8) compared to that measured in the ambient water (meanwat = 3.9 ± 1.4) (Table 2). These results clearly contrast with the refuge theory that predicts rather low prevalence of lysogeny in highly ‘bacterialized’ habitats, like the fish gut. Interestingly, such high proportions of lysogens were also found in the coral mucus which is a nutrient-rich medium, and which typically hosts an abundant community of bacteria27,28. Recently a new model, the piggyback-the-winner model postulates that lysogeny can dominate in certain biomes hosting abundant and fast growing bacteria29. This may be the case in animal microbiomes where commensal bacteria are abundant, functionally crucial for their host, and to which lysogeny could represent an advantageous strategy, providing cells immunity against viral surinfections together with acquisition of genes allowing optimal survival in stressing conditions30,31. Temperate phages have also been long shown to prevail in the human gut9,32, suggesting that the digestive tract may harbor a cosmopolitan lysogenic phageome capable of controlling the bacterial homeostastis in vertebrates at large. The metagenomic research of prophages sequences in a large variety of terrestrial and marine vertebrates will be necessary to further explore this sensitive question.

We did not observe any significant correlation between salinity and any of the studied microbial parameters. Overall, given the euryhaline nature of S. melanotheron heudelotii12 and because the fish intestine is a major osmoregulatory organ, it was not surprising to see that gut-associated bacterial and viral communities were not impacted by external osmolarity.

One of the most striking findings here was the positive correlation between the fraction of lysogenized cells and the concentrations of polycyclic aromatic hydrocarbons (PAH). We know that the gut bacteriome of fishes (and of human) is affected by external environmental conditions33, including the presence of xenobiotics34,35. However the potential relationships between gut-associated viruses and environmental pollutants have only received little attention so far. PAHs are ubiquitous environmental pollutants, generated primarily during the incomplete combustion of organic materials (e.g. coal, oil, petrol, and wood), which exhibit toxic, mutagenic and/or carcinogenic properties to many aquatic organisms36,37. Previous reports have shown that PAHs can accumulate into the sediment38, and since they are highly lipid soluble, they can be readily absorbed from the intestinal tract of marine vertebrates39. Herein, the positive relationship between the PAH concentrations and FLC may indicate that phage-bacteria interactions within the fish gut are also strongly susceptible to this specific persistent pollutants, contrary the other POPs investigated here (e.g. PCB and OCP). Since the lysis-lysogenic decision usually depends upon host cell metabolism40, one may suspect that the physiological stress caused by the presence of PAH could have triggered lysogenic pathways. This may impact the fragile balance between temperate and virulent phages within the gut microbiota, with potentially severe implications for fish health.


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