160 CHAPTER Q2 DESIGN FOR PERFORMANCE PC D/A amp motor beam A/D filter Figure 10.1:Flexible beam setup. beam has an infinite number of modes.The model is therefore linear but infinite-dimensional.The corresponding transfer function from torque input at the motor end to tip deflection at the sensed end has the form 00 + i=0 Then damping was introduced,yielding the form 00 s2 +2Giwis +w? =0 The first term is co/s2 and corresponds to the rigid-body slewing motion about the pinned end. The second term, C- 82+2(w-8+w2 corresponds to the first flexible mode.And so on.The motion was found to be adequately modeled by the first four flexible modes.Then the damping rat ios and natural frequencies were determined experimentally.Finally,the amplifier,motor,and sensor were introduced into the model.The antialiasing filter was ignored for the purpose of design. For simplicity we shall take the plant transfer function to be -6.4750s2+4.0002s+175.7700 P(8)=35s+Q5682s2+109.50218+0.0929 The poles are 0,-0.0007,-0.0565±5.2700j. The first two poles correspond to the rigid-body motion;the one at s=-0.0007 has been perturbed away from the origin by the back EMF in the motor.The two complex poles correspond to the first flexible mode,the damping ratio being 0.0675.The zeros are -4.9081,5.5008. Because of the zero at s =5.5008 the plant is non-minimum phase,reflecting the fact that the actuator (the motor)and the sensor are not located at the same point on the beam.The procedure of the preceding section requires no poles on the imaginary axis,so the model is (harmlessly) perturbed to -6.4750s2+4.0002s+175.7700 P(8)=53+0.5682s9+109.502152+0.0929s+10 CHAPTER  DESIGN FOR PERFORMANCE PC DA amp motor beam AD lter sensor ￾ ￾ ￾ ￾ ￾ Figure  Flexible beam setup beam has an innite number of modes The model is therefore linear but innite dimensional The corresponding transfer function from torque input at the motor end to tip deection at the sensed end has the form X￾ i ci s   i  Then damping was introduced yielding the form X￾ i ci s   iis   i  The rst term is cs and corresponds to the rigid body slewing motion about the pinned end The second term c￾ s   ￾￾s   ￾ corresponds to the rst exible mode And so on The motion was found to be adequately modeled by the rst four exible modes Then the damping ratios and natural frequencies were determined experimentally Finally the amplier motor and sensor were introduced into the model The antialiasing lter was ignored for the purpose of design For simplicity we shall take the plant transfer function to be P s   s   s   ss  s   s     The poles are     j The rst two poles correspond to the rigid body motion the one at s    has been perturbed away from the origin by the back EMF in the motor The two complex poles correspond to the rst exible mode the damping ratio being   The zeros are     Because of the zero at s    the plant is non minimum phase reecting the fact that the actuator the motor and the sensor are not located at the same point on the beam The procedure of the preceding section requires no poles on the imaginary axis so the model is harmlessly perturbed to P s   s   s   s  s   s   s