ME 451: Control Systems Laboratory Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 1 Department of Mechanical Engineering Michigan State University East Lansing, MI 48824-1226 ME451 Laboratory Experiment #5 Sinusoidal Response of a 2nd Order Torsional Mass-Spring Damper System Important: Bring your short form from lab #2 to the lab. You will need them for comparison. __________________ ME451 Laboratory Manual Pages, Last Revised: October 31, 2007 Send comments to: Dr. Clark Radcliffe, ProfessorME 451: Control Systems Laboratory Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 2 Reference: C.L. Phillips and R.D. Harbor, Feedback Control Systems, Prentice Hall, 4th Ed. Section 4.2, pp. 121-124: Time Response of Second-Order Systems Section 4.4, pp. 129-132: Frequency Response of Systems Appendix B, pp. 635-650: Laplace Transform 1. Objective The response of a linear system to a sinusoidal input is useful for predicting its behavior for arbitrary periodic inputs, but more importantly, for compensator design. For second-order systems, the sinusoidal response depends primarily on the natural frequency, ωn, and the damping ratio, ζ. Both ωn and ζ are functions of system parameters, both physical and control parameters. In this experiment we will alter the ωn and ζ values by changing the feedback control gains. The objective of this experiment is to investigate the relationship between ωn and ζ and the frequency response of the system, as well as the relationship between the feedback gains and ωn and ζ. The second-order system we choose for this experiment is a torsional mass-spring-damper system, with torque as input and angular displacement as output. We obtain the transfer function of the system and identify specific parameters of the system that affect sinusoidal response. Specifically, we identify parameters that affect the natural frequency and the damping ratio. We vary these parameters to experimentally verify the change in sinusoidal response. 2. Background 2.1. Second-order systems The standard form of transfer function of a second-order system is 2222)()()(nnnssKsUsYsGωςωω++== (1) where Y (s) and U(s) are the Laplace transforms of the output and input variables, respectively, ωn is the natural frequency, and ζ is the damping ratio. For a sinusoidal input )*sin()( tAtuω=, 22)(ωω+=sAsU the response of the system, in Laplace domain, can be written as )2)(()(22222nnnsssKAsYωςωωωω+++= Assuming poles of G(s) are in the left-half plane, the steady state response of the system (after transients have decayed) can be written as )*sin()()(φωω+= tjGAty , )(ωφjG∠≡ (2) It is clear from Eq.(2) that a sinusoidal input produces a sinusoidal output. The amplitude of the output is scaled by a factor of )(ωjG and the phase lags behind the input by )(ωjG∠.ME 451: Control Systems Laboratory Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 3 For the standard second-order system in Eq.(1), given the values of ωn and ζ, the “gain" )(ωjG and the “phase" )(ωjG∠ can be expressed as a function of ω, as follows 22224)1()(rrKjGςω+−= ; )(nrωω≡ (3) −−=∠=21*2arctan)(rrjGςωφ (gives units of radians) On a logarithmic scale, they can be plotted to generate what are known as gain and phase plots, or Frequency Response diagrams. The Frequency Response diagrams for a standard second-order system are plotted as a function of the frequency ratio )(nrωω≡ , for different values of ζ in Fig.1. Figure 1. Frequency Response diagram for standard second-order system 2.2. Torsional mass-spring-damper system Recall the torsional mass-spring-damper system in laboratory experiment #2, shown here again in Fig.2. The system variables are T external torque applied on rotor θ angular position of rotor ω angular velocity of rotor The parameters of the system, shown in Fig.2, includeME 451: Control Systems Laboratory Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 4 J moment of inertia of rotor b coefficient of viscous friction k spring constant The transfer function of the mass-spring-damper system, with T as input and θ as output, can be written as ++=++==)/()/()/(11)()()(22JksJbsJkkkbsJssTssGθ (4) Figure 2. Torsional mass-spring-damper system The transfer function above bears close resemblance with the standard second-order transfer function in Eq.(1). The only difference is the DC gain of (1/k), which appears in Eq.(4). By comparing Eqs.(1) and (4), the expressions for natural frequency and damping ratio can be obtained as Jkn=ω, Jbn=ζω2 , Jkb2=ζ (5) It is clear from Eq.(5) that ωn and ζ are functions of system parameters J, b, and k, which are typically fixed. Hence, if we build the system in Fig.2, we will have a specific second-order response. For studying the effect of system parameters on the response, we must be able to change J, b, and k. We achieve this by removing the spring and damper in Fig.2 and programming a motor to generate the torques generated by the spring and the damper. Specifically, the motor is programmed to generate the torque given by the relation )(21θθθ&KKKKkTdpfhe−−= ** If you had calculated the gain ek from earlier experiment you would remember that it was found to be close to 25ME 451: Control Systems Laboratory Sinusoidal Response of a Second Order Plant: Torsional Mass-Spring Damper System 5 Figure 3. Block Diagram of Programmable Torsional mass-spring-damper system where hK is the hardware gain of the motor; T is the input torque; the feedback generated torqueθ1K− is the type of torque generated by a spring; and the feedback generated torque .2θK is the type of torque generated by a damper. When this torque is applied on the rotor, the dynamic equation of the programmed system becomes TkbJ =++θθθ.&& ⇒ θθθθθθ12KkKKkKKkKkbJehehdpfeh−−=++&&&& and then ()()()dpfehehehKkKKkKkKkKbJθθθθ=−+−+12&&& The block diagram of the programmed system is shown in Fig.3. The transfer function of this system, with dθ as input and θ as output, can then be expressed as ( )[ ]( )[ ]++++==JKkKksJKkKbsJKkKsssGehehpfehd//)/()()()(122θθ (6) which has
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