Massachusetts Institute of Technology Department of Mechanical Engineering 2.003J/1.053J Dynamics & Control I Fall 2007 Homework 6 Issued: Oct. 19. 2007 Due: Nov. 2. 2007 Problem 6.1 : Dynamics of a mass-spring-damper system Ft= F sin ωt k xmb x0& v0 ( ) 0 ( ) A mass-spring-damper system is subject to an external sinusoidal force Ft() with amplitude F0, and angular frequency ω. The position of particle, x, is zero when the spring is neither compressed or stretched. The mass of the particle is m , the damping coefficient is b and the spring constant is k . Initial condition is expressed as x(0) = x0 and v(0) = v0 . Derive the equation of motion for this mass-spring-damper system. Problem 6.2 : Solver for a mass-spring-damper system using Euler method You are familiar with Euler method used to obtain the trajectory of a ball dropping and bouncing in Homework #3. In this homework, you will create a function that use Euler method to derive the particle trajectory (two column matrix for time and position each) in mass-spring-damper system from t=0 to 50 sec. The input parameters of the function are coefficients ( m , b , k , F0and ω) and initial conditions ( x0 and 0Use time increment Δ = v ). t 0.1sec . Function name (and m-file name) should be ‘MSDSE_your_kerberos_name’ and upload it to 2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments. Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Problem 6.3 : Solver for a mass-spring-damper system with using Runge-Kutta method Using Runge-Kutta algorithm to solve the differential equation of this system, generate a m-file function with input parameters ( m , b , k , F0, ω, x0 and v0 ) to calculate particle trajectory (two column matrix for time and position each) from t=0 to 50 sec. You may use either Runge-Kutta 23 or 45 algorithm (See ode23 or ode45 in the MATLAB help, and use one of them in your m-code.). Use t 0.1sec Δ= for both the time step size for the solver (See also odeset in the MATLAB help) and the time step for the output time from the solver (See input argument for tspan in ode23 or ode45). Function name (and m-file name) should be ‘MSDSRK_your_kerberos_name’ and upload it to 2.003 MIT Server site. You also submit the hardcopy of your code with appropriate comments. Problem 6.4 : Trajectory of a mass-spring-damper system with different parameters and initial conditions For the following cases, plot the time response of the particle. Plot the trajectories with Euler and Runge-Kutta solvers on a single graph with appropriate legends. Compare the results for these two approaches. Submit a hardcopy of your trajectory plots. i) m = 1kg , b = ⋅ , k = 1N/m , F0 = 1N , ω= 3rad/sec , x(0) =0.5 N sec/m 0m and v(0) = 0m /sec ii) m =1kg , b = ⋅ , k = 1 N/m , F0 = 0N , ω= 0 rad/sec , x(0) = m and0.5 N sec/m 1 v(0) = 0m /sec iii) m = 1kg , b = ⋅ , k = 1 N/m , F0 = 1N , ω= 1 rad/sec , x(0) = 0m and0 N sec/m v(0) = 0m /sec Cite as: Peter So, course materials for 2.003J / 1.053J Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month
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