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Impulse vector

Impulse vector is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Impulse vector rather than just read about it. In short: An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified manner.

Impulse vector — main illustration
Impulse vector — illustration

Key takeaways

  • Impulse vector belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Impulse vector to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Impulse vector from memory before moving on to harder problems.

Reference excerpt

An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified manner. The impulse vector makes it easy to obtain impulse time and magnitude of the input shaper graphically. A vector concept for an input shaper was first introduced by W. Singhose for undamped systems with positive impulses. Building on this idea, C.-G. Kang introduced the impulse vector (or Kang vector) to generalize Singhose's idea to undamped and underdamped systems with positive and negative impulses.

Definition

For a vibratory second-order system ω n 2 / ( s 2 + 2 ζ ω n + ω n 2 ) {\displaystyle \omega _{n}^{2}/(s^{2}+2\zeta \omega _{n}+\omega _{n}^{2})} with undamped natural frequency ω n {\displaystyle \omega _{n}} and damping ratio ζ {\displaystyle \zeta } , the magnitude I i {\displaystyle I_{i}} and angle θ i {\displaystyle \theta _{i}} of an impulse vector (or Kang vector) I i {\displaystyle \mathbf {I} _{i}} corresponding to an impulse function A i δ ( t − t i ) {\displaystyle A_{i}\delta (t-t_{i})} , i = 1 , 2 , . . . , n {\displaystyle i=1,2,...,n} is defined in a 2-dimensional polar coordinate system as

I i = A i e ζ ω n t i {\displaystyle I_{i}=A_{i}e^{\zeta \omega _{n}t_{i}}}

θ i = ω d t i {\displaystyle \theta _{i}=\omega _{d}t_{i}}

… excerpt ends here. Continue reading the full article.

Illustrations

Impulse vector: Two impulse vectors and their corresponding impulse responses for a second-order system. (a) Two impulse vectors I 1 , I 2 {\displaystyle \mathbf {I} _{1},\mathbf {I} _{2}} with the same magnitude and 180 deg angle difference, with one pointing to the origin and the other pointing to the outside, are regarded as the same vector for vector addition and subtraction. (b) Two impulse responses corresponding to two impulse vectors are exactly same after the final impulse time t 2 {\displaystyle t_{2}}
Two impulse vectors and their corresponding impulse responses for a second-order system. (a) Two impulse vectors I 1 , I 2 {\displaystyle \mathbf {I} _{1},\mathbf {I} _{2}} with the same magnitude and 180 deg angle difference, with one pointing to the origin and the other pointing to the outside, are regarded as the same vector for vector addition and subtraction. (b) Two impulse responses corresponding to two impulse vectors are exactly same after the final impulse time t 2 {\displaystyle t_{2}}
Impulse vector: (a) Two representations I R 1 , I R 2 {\displaystyle \mathbf {I} _{R1},\mathbf {I} _{R2}} of the resultant of two impulse vectors I 1 , I 2 {\displaystyle \mathbf {I} _{1},\mathbf {I} _{2}} , and (b) the corresponding impulse responses.
(a) Two representations I R 1 , I R 2 {\displaystyle \mathbf {I} _{R1},\mathbf {I} _{R2}} of the resultant of two impulse vectors I 1 , I 2 {\displaystyle \mathbf {I} _{1},\mathbf {I} _{2}} , and (b) the corresponding impulse responses.
Impulse vector: Cancelling impulse vector I 3 {\displaystyle \mathbf {I} _{3}} causes no residual vibration after the final impulse time t 3 {\displaystyle t_{3}} when the impulse sequence A 1 δ ( t ) + A 2 δ ( t − t 2 ) + A 3 δ ( t − t 3 ) {\displaystyle A_{1}\delta (t)+A_{2}\delta (t-t_{2})+A_{3}\delta (t-t_{3})} is applied to an undamped or underdamped second-order system because I 1 + I 2 + I 3 = 0 {\displaystyle \mathbf {I} _{1}+\mathbf {I} _{2}+\mathbf {I} _{3}=\mathbf {0} } .
Cancelling impulse vector I 3 {\displaystyle \mathbf {I} _{3}} causes no residual vibration after the final impulse time t 3 {\displaystyle t_{3}} when the impulse sequence A 1 δ ( t ) + A 2 δ ( t − t 2 ) + A 3 δ ( t − t 3 ) {\displaystyle A_{1}\delta (t)+A_{2}\delta (t-t_{2})+A_{3}\delta (t-t_{3})} is applied to an undamped or underdamped second-order system because I 1 + I 2 + I 3 = 0 {\displaystyle \mathbf {I} _{1}+\mathbf {I} _{2}+\mathbf {I} _{3}=\mathbf {0} } .
Impulse vector: Impulse vector diagrams for (a) ZV, (b) ZVD, (c) ZVD2, and (d) ZVD3 shapers.
Impulse vector diagrams for (a) ZV, (b) ZVD, (c) ZVD2, and (d) ZVD3 shapers.
Impulse vector: Impulse vector diagram of (a) an ETM4 shaper, (b) ETM5 shaper, and (c) ETM6 shaper.
Impulse vector diagram of (a) an ETM4 shaper, (b) ETM5 shaper, and (c) ETM6 shaper.

Worked examples

Example 1 — a first encounter with Impulse vector

Start with the simplest possible case. Write down what Impulse vector claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Impulse vector before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Impulse vector ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Impulse vector

In research
Impulse vector appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Impulse vector in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Impulse vector is common in secondary-school and first-year university syllabi. It links to neighbouring topics Control theory, Dynamics (mechanics), Mechanical vibrations, so understanding it makes those chapters shorter.
In everyday life
Look for Impulse vector outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Impulse vector in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Impulse vector means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Impulse vector out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Impulse vector in simple terms?

An impulse vector, also known as Kang vector, is a mathematical tool used to graphically design and analyze input shapers that can suppress residual vibration. The impulse vector can be applied to both undamped and underdamped systems, as well as to both positive and negative impulses in a unified…

Why does Impulse vector matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Impulse vector?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Impulse vector.

Tags

  • Control theory
  • Dynamics (mechanics)
  • Mechanical vibrations

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