1 ME 451: Control Systems Laboratory Department of Mechanical Engineering Michigan State University East Lansing, MI 48824-1226 ME451 Laboratory Experiment #4 Sinusoidal Response of a First Order Plant DC Servo Motor *** Please bring your Experiment #1 Short Form for this lab*** ______________________________________ ME451 Laboratory Manual Pages, Last Revised: March 31, 2008 Send comments to: Dr. Clark Radcliffe, ProfessorExpt #3, Sinusoidal Response of a First Order Plant: DC Servo Motor_________________________ 1 References: C.L. Phillips and R.D. Harbor, Feedback Control Systems, Prentice Hall, 4th Ed. Section 2.7, pp. 38-43: Electromechanical Systems Section 4.1, pp. 116-120: Time Response of First Order Systems Section 4.4, pp. 129-132: Frequency Response of Systems Section 8.1, 8.2, pp. 275-293: Frequency Responses, Bode Diagrams 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 first-order systems, the sinusoidal response depends on the DC gain, K, but primarily on the time constant, τ. Both K and τ are functions of system parameters. The objective of this experiment is to investigate the effect of system parameters on system response to a sinusoidal input. We will experiment with an armature controlled DC servo motor, which we approximate will behave as a first-order system with voltage as the input and angular velocity as the output. The transfer function of the system will be obtained to identify specific parameters that affect sinusoidal response. We seek primarily the system parameters which affect the gain and time constant. We will vary these parameters to experimentally verify the change in the sinusoidal response. 2. Background 2.1. Sinusoidal Command: A sine wave: is characterized by 3 parameters: amplitude CA , period T (seconds) and phase φ (radians) Figure 1. Sine Wave Parameters.Expt #3, Sinusoidal Response of a First Order Plant: DC Servo Motor_________________________ 2 The phase φ (radians) of a sine wave is a relative quantity; since the sine function can take any argument and has no absolute starting point. Mathematically, a sine wave y(t) varying with time t is described by: )sin()(φω+= tAtyc (1) Here Ac is the amplitude and can have any units (feet, volts, psi, etc.) the quantity y(t) represents in its physical form. The phase φ has units of angle, either degrees or radians. In the equation as written ω has a unit of angle per unit time, because the quantity ωt must have units of angle. The angular frequency ω (rad/sec) is 2π times the circular frequency f (cycles/sec = Hertz). The circular frequency, f (Hz) is the inverse of the wave period (seconds/cycle). Tfππω22 == (2) 2.2. Sinusoidal response: Consider a linear process with a sinusoidal input, whose output is observed. Below we examine the output and input to observe potential defferences in the signals. ∆t2Ac2ArCommandResponse∆t2Ac2ArCommandResponse Figure 2. Sinusoidal Command versus Sinusoidal Response. • The output will be a sinusoid of the same frequency as the input. • The ratio of the output amplitude to that of the input amplitude (often called process gain) will in general vary with the frequency of the sine wave input. • The difference in phase between the input and output sine waves will also depend on the frequency.Expt #3, Sinusoidal Response of a First Order Plant: DC Servo Motor_________________________ 3 We call the ratio of the output amplitude to input amplitude for any given time the process Gain. This is a function of time, as the gain changes throughout the wave if there is any phase angle (described below with eq. (4) ). )sin()sin()()()(CCCRRRCRtAtAtytyjGφωφωω++== (3) The gain amplitude is defined as CRAAjG =)(ωand the difference between input and output phases is the phase angle ( )(degrees) 360(radians) 2)( tTtTjGCR∆=∆=−=∠πφφω (4) 2.3. Frequency response of first-order systems to sinusoidal commands: The standard form of transfer function of a first-order system is 1)()()(+==sKsUsYsGτ (5) where Y (s) and U(s) are the Laplace transforms of the output and input variables, respectively, K is the DC gain, and τ is the time constant. For a sinusoidal input 22)(),sin()(ωωω+==sAsUtAtu (6) The response of the system, in Laplace domain, can be written as )1)(()(22++=ssKAsYτωω (7) 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()()(ωφφωωjGtjGAty ∠=+=∆ (8) It is clear from (8) 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∠. For the standard first-order system in (5), given the values of K and τ, the “gain” )(ωjG and the “phase” )(ωjG∠ can be expressed as a function of ω, )arctan()(1)(22ωτωτωω−=∠+= jGKjG (9) For a process we may plot )(ωjG and )(ωjG∠ as functions of frequency together in what is called a Frequency Response Diagram. Amplitude ratio )(ωjG may be thought of as the frequency dependent process gain. It may be expressed in dimensional or dimensionless form.Expt #3, Sinusoidal Response of a First Order Plant: DC Servo Motor_________________________ 4 The latter is preferred. In practice the interesting range of )(ωjG may cover several orders of magnitude. For this reason it is often expressed on a logarithmic scale in decibels (dB). This can only be used when the )(ωjG has been made dimensionless, and the relevant definition is: ())(20)(10ωjGLogdBGain = (10) On a logarithmic scale the open loop gains)(ωjG and )(ωjG∠ can be plotted to generate what are known as gain and phase plots, or Bode diagrams. The frequency response diagrams for a standard first-order system are shown in Fig.3. Notice on these diagrams that for a first order system, there are no peaks in gain, the gain roll off (slope as ω goes to ∞) is always 20 dB/dec, and the phase lag does not drop below -90°. Figure 3. Frequency response of a first-order system to a sinusoidal command. 2.4. DC servo motor system Recall the armature controlled DC servo motor in laboratory experiment #1 and shown here again in Fig.4. The
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