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Class 4 Dynamic Performance

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The dynamic characteristics of a measuring instrument describe its behavior between the time a measured quantity changes value and the time when the instrument output attains a steady value in response. Because dynamic signals vary with time, the measurement system must be able to respond fast enough to keep up with the input signal. Further, we need to understand how the input signal is applied to the sensor because that plays a role in system response.

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Mechanical Zero-Order Systems The simplest model of a measurement system is the zero-order system model. This is represented by the zero-order differential equation: K is the static sensitivity or steady gain of the system. It is a measure of the amount of change in the output in response to the change in the input. l1l1 l2l2 f(t) x(t) x(t)/l 2 = f(t)/l 1 K = l 2 /l 1

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Mechanical Zero-Order Systems In a zero-order system, the output responds to the input signal instantaneously. If an input signal of magnitude f(t) = A were applied, the instrument would indicate KA. The scale of the measuring device would normally be calibrated to indicate A directly. l1l1 l2l2 f(t) x(t) x(t)/l 2 = f(t)/l 1 K = l 2 /l 1

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Electrical Zero-Order Systems In a zero-order system, the output responds to the input signal instantaneously. vivi R1R1 R2R2 VoVo

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A Unity Gain Zero-Order system l xixi xoxo x o (t) = x i (t) K = 1

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Non-zero Order Systems Measurement systems that contain storage or dissipative elements do not respond instantaneously to changes in input. In the bulb thermometer, when the ambient temperature changes, the liquid inside the bulb will need to store a certain amount of energy in order for it to reach the temperature of the environment. The temperature of the bulb sensor changes with time until this equilibrium is reached, which accounts physically for its lag in response. In general, systems with a storage or dissipative capability but negligible inertial forces may be modeled using a first-order differential equation. m k c xixi xoxo

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Non-zero Order Systems Measurement systems that contain storage or dissipative elements do not respond instantaneously to changes in input. In the bulb thermometer, for example, when the ambient temperature changes, the liquid inside the bulb will need to store a certain amount of energy in order for it to reach the temperature of the environment. The temperature of the bulb sensor changes with time until this equilibrium is reached, which accounts physically for its lag in response.

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First Order Systems Consider the time response of a bulb thermometer for measuring body temperature. The thermometer, initially at room temperature, is placed under the tongue. Body temperature itself is constant (static) during the measurement, but the input signal to the thermometer is suddenly changed from room temperature to body temperature. This, is a step change in the measured signal. The thermometer must gain energy from its new environment to reach thermal equilibrium, and this takes a finite amount of time. The ability of any measurement system to follow dynamic signals is a characteristic of the measuring system components. Body Temperature Room Temperature t time

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First Order Systems Suppose a bulb thermometer originally at temperature T o is suddenly exposed to a fluid temperature T ∞. Develop a model that simulates the thermometer output response. Rate of energy stored = Rate of energy in The ratio mcp/hA has a units of seconds and is called the time constant Body Temperature Room Temperature t time

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First Order Systems The ratio mcp/hA has a units of seconds and is called the time constant, τ. If the time constant is much less then 1 second, the system may be approximated by a unity gain zero-order system. Body Temperature Room Temperature t time

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1 st Order Systems Examples: Bulb Thermometer RC Circuits Terminal velocity Mathematical Model:

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1 st Order Systems with Step Input

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Second Order Systems In the system shown, the input displacement, x i, will cause a deflection in the spring, and some time will be needed for the output displacement x o to reach the input displacement. m k c xixi xoxo

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Second Order Systems If m/k << 1 s 2 and c/k << 1 s, the system may be approximated as a zero order system with unity gain. If, on the other hand, m/k << 1 s 2, but c/k is not, the system may be approximated by a first order system. Systems with a storage and dissipative capability but negligible inertial may be modeled using a first-order differential equation. m k c xixi xoxo

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Example – Automobile Accelerometer Consider the accelerometer used in seismic and vibration engineering to determine the motion of large bodies to which the accelerometer is attached. The acceleration of the large body places the piezoelectric crystal into compression or tension, causing a surface charge to develop on the crystal. The charge is proportional to the motion. As the large body moves, the mass of the accelerometer will move with an inertial response. The stiffness of the spring, k, provides a restoring force to move the accelerometer mass back to equilibrium while internal frictional damping, c, opposes any displacement away from equilibrium. m k c xixi xoxo Piezoelectric crystal

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Zero-Order systems Can we model the system below as a zero-order system? If the mass, stiffness, and damping coefficient satisfy certain conditions, we may. m k c xixi xoxo

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First Order Systems Measurement systems that contain storage elements do not respond instantaneously to changes in input. The bulb thermometer is a good example. When the ambient temperature changes, the liquid inside the bulb will need to store a certain amount of energy in order for it to reach the temperature of the environment. The temperature of the bulb sensor changes with time until this equilibrium is reached, which accounts physically for its lag in response. In general, systems with a storage or dissipative capability but negligible inertial forces may be modeled using a first-order differential equation.

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1 st Order Systems with Step Input Error Ratio Excitation Ratio Note that the excitation ratio also represents the system response in case of x0=0 and K=1 Excitation ratio may also be called response ratio = current response / desired response

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Example 1 A bulb thermometer with a time constant τ =100 s. is subjected to a step change in the input temperature. Find the time needed for the response ratio to reach 90%

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Example 1 Solution A bulb thermometer with a time constant τ =100 s. is subjected to a step change in the input temperature. Find the time needed for the response ratio to reach 90%

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1 st Order Systems with Ramp Input

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1 st Order Systems with Harmonic Input

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The amplitude ratio, Ar(ω), and the corresponding phase shift, ϕ, are plotted vs. ωτ. The effects of τ and ω on frequency response are shown. For those values of ωτ for which the system responds with Ar near unity, the measurement system transfers all or nearly all of the input signal amplitude to the output and with very little time delay; that is, X will be nearly equal to F in magnitude and ϕ will be near zero degrees.

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1 st Order Systems with Harmonic Input At large values of ωτ the measurement system filters out any frequency information of the input signal by responding with very small amplitudes, which is seen by the small Ar(ω), and by large time delays, as evidenced by increasingly nonzero ϕ.

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1 st Order Systems with Harmonic Input Any equal product of ω and τ produces the same results. If we wanted to measure signals with high- frequency content, then we would need a system having a small τ. On the other hand, systems of large τ may be adequate to measure signals of low-frequency content. Often the trade-offs compete available technology against cost. dB = 20 log Ar(ω)

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1 st Order Systems with Harmonic Input The dynamic error,δ(ω), of a system is defined as δ(ω) = (X(ω) – F)/F δ(ω) = Ar(ω) –1 It is a measure of the inability of a system to adequately reconstruct the amplitude of the input signal for a particular input frequency. We normally want measurement systems to have an amplitude ratio at or near unity over the anticipated frequency band of the input signal to minimize δ(ω). As perfect reproduction of the input signal is not possible, some dynamic error is inevitable. We need some way to quantify this. For a first-order system, we define a frequency bandwidth as the frequency band over which Ar(ω) > 0.707; in terms of the decibel defined as dB = 20 log Ar(ω) This is the band of frequencies within which Ar(ω) remains above 3 dB

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Example 2 A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that a dynamic error of 2% is acceptable.

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Example 2. Solution A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that an absolute value for the dynamic error of 2% is acceptable. Accordingly, a sensor having a time constant of 6.4 ms or less will work.

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Example 2. Solution A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected that the temperature will behave as a simple periodic waveform with a frequency somewhere between 1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time constant, select a suitable sensor, assuming that an absolute value for the dynamic error of 2% is acceptable. Accordingly, a sensor having a time constant of 6.4 ms or less will work.

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2 nd Order Systems Example: Spring – mass damper RLC Circuits Accelerometers Mathematical Model:

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2 nd Order Systems with step input

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Correction to Figliola’s Book Theory and Design for Mechanical Measurements The sign should be – not +

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2 nd Order Systems with step input

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2 nd Order Systems with periodic input

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2 nd Order Systems with step input

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