Mechanical Systems: Forces and Transfer Functions
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What is the correct expression for the transfer function from u(s) to x1(s)?

  • G1 = [K1 + K2] / [m1s² + bs]
  • G1 = (K2 + bs) / [m1s² + bs + K1]
  • G1 = m1 / [s² + bs + K1 + K2]
  • G1 = 1 / [m1s² + bs + K1 + K2] (correct)

    • Which equation correctly represents the forces acting on mass m2?

  • K2(x1-x2) + b(x1-x2) - K3x2 = m2x2 (correct)
  • K2(x1-x2) + b(x1-x2) = m2s²x2(s)
  • K1(x1-x2) + b2(x1-x2) = m2s²x2(s)
  • K2(x1-x2) + K3x2 = m2x2

    • Which expression correctly gives the final transfer function T.F?

  • T.F = (K2 + bs) / [(m1s² + bs + K1 + K2)(m2s² + bs + K2 + K3) - (K2 + bs)²] (correct)
  • T.F = G1G2 / (1 - G1G2H)
  • T.F = G1G2 / (1 + G1G2H)
  • T.F = (K2 + bs) / [(m1s² + bs)(m2s² + K3)]

    • What represents the forces acting on mass m1 according to Newton's second law?

    <p>F(t) - K1(x-y) - b1(x-y) = m1.s²x1(s)</p> Signup and view all the answers

    Which statement is true regarding the simplification of the equation for mass m2?

    <p>x1(s)[K2 + bs] = x2(s) [m2s² + K2 + K3]</p> Signup and view all the answers

    What does the variable K represent in the spring force equation F(t) = K x(t)?

    <p>Stiffness factor</p> Signup and view all the answers

    In the Laplace domain, how is the force generated by friction defined?

    <p>F(s) = B s x(s)</p> Signup and view all the answers

    Which equation represents the transfer function (T.F.) in a transitional system?

    <p>T.F. = x(s) / F(s)</p> Signup and view all the answers

    What is the result of applying Newton's Second Law in a mass-spring-damper system?

    <p>F(t) = m a</p> Signup and view all the answers

    How is the input force represented in the force equation for mass m1 of a two-mass system?

    <p>F(t) - K1(x-y) - K2(y-z) - B1y = m1.ÿ</p> Signup and view all the answers

    What does the equation F(s) = x(s) [ms² + BS + K] imply about the relationship between force and displacement in the Laplace domain?

    <p>It represents a second-order dynamic system.</p> Signup and view all the answers

    When analyzing a damped system, what role does the damping constant B play in the force equation F(t) = Bx(t)?

    <p>It affects the velocity of the system.</p> Signup and view all the answers

    What is the purpose of a free body diagram in the context of mechanical systems?

    <p>To show the forces acting on an object.</p> Signup and view all the answers

    What is the general form of the transfer function from x(s) to y(s) as derived from the system?

    <p>G1 = K1 / [m1s² + B1s + K2 + K1]</p> Signup and view all the answers

    Which equation correctly represents the forces acting on mass m2?

    <p>K1(x - y) - K2y - B2s*y = m2s²y</p> Signup and view all the answers

    What expression describes the final transfer function T.F for the system?

    <p>T.F = G1G2 / (1 - G1G2H)</p> Signup and view all the answers

    In the context of the Laplace Transform, what does the variable s commonly represent?

    <p>Frequency domain variable</p> Signup and view all the answers

    From the equation K2y(s) = z(s) [m2s² + B2s + K3 + K2], what does z(s) represent?

    <p>The output displacement of mass m2</p> Signup and view all the answers

    Which relationship correctly expresses the connection between input force F(s) and output x(s) for mass m1?

    <p>F(s) - K1(x - y) - B1sx = m1s²x</p> Signup and view all the answers

    What is the coefficient of the damping factor within the equation for mass m2?

    <p>B2</p> Signup and view all the answers

    Which of the following is NOT a component of the transfer function from y(s) to z(s)?

    <p>K1</p> Signup and view all the answers

    What is the form of the transfer function from x(s) to y(s)?

    <p>$G1 = \frac{1}{[m1s² + b1s + K1]}$</p> Signup and view all the answers

    Which equation represents the forces acting on mass m2?

    <p>$u2 + b(y1 - y2) - K2y2 = m2.ÿ2$</p> Signup and view all the answers

    What is the final form of the transfer function (T.F.) given in the content?

    <p>$T.F = \frac{K1 + b1s}{(m1s² + b1s + K1)(m2s² + b2S + b1s + K1 + K2) - (K1 + b1s)²}$</p> Signup and view all the answers

    In the system described, what happens to the transfer function from u1(s) to y2(s) when u2(s) is set to 0?

    <p>It simplifies to only the response from mass m1.</p> Signup and view all the answers

    What is the equation for finding the relationship between x(s) and y(s) represented in terms of the given parameters?

    <p>$x(s)[K1 + b1s] = y(s)[m2s² + b2S + b1s + K1 + K2]$</p> Signup and view all the answers

    What is the purpose of applying the Laplace transform in the analysis of the system?

    <p>To convert time-domain differential equations to algebraic equations.</p> Signup and view all the answers

    What values do G1, G2, and G3 represent in the context of the transfer function calculations?

    <p>The transfer functions relating input forces to outputs.</p> Signup and view all the answers

    How is the final transfer function derived from G1, G2, and G3?

    <p>By multiplying the individual transfer functions and applying feedback.</p> Signup and view all the answers

    Study Notes

    System Components

    • Springs generate force proportional to displacement: F(t) = K x(t).
    • Friction generates force proportional to velocity: F(t) = B x(t).
    • Dampers generate force proportional to velocity: F(t) = Bx(t).

    Transitional Systems

    • Transitional systems aim to find the relationship between input and output.
    • The Transfer Function (T.F.) represents this relationship: T.F.= x(s)/F(s).
    • Newton's Second Law states that the sum of forces equals mass times acceleration: ΣF = m.a.

    Example 1: Single Mass System

    • A single mass m is connected to a spring with stiffness K and a damper with damping constant B.
    • An external force F(t) is applied to the mass.
    • The Transfer Function for this system is: T.F = x(s)/F(s) = 1/(ms² + BS + K).

    Example 2: Single Mass System (Simplified)

    • This example is identical to Example 1, highlighting the same key concepts.
    • The Transfer Function is identical: T.F = x(s)/F(s) = 1/(ms² + BS + K).

    Example 3: Two Mass System

    • Two masses m1 and m2 are connected by two springs K1, K2 and two dampers B1, B2.
    • An external force F(t) is applied to the system.
    • The output is the displacement of the second mass x(t).
    • This system has two masses, requiring separate equations for force balance.
    • The Transfer Function is complex due to the interactions between the two masses and the springs and dampers.

    Example 4: Two Mass System (Modified)

    • Similar to Example 3, this system features two masses m1 and m2 connected by springs and dampers.
    • However, the external force is applied to the first mass m1.
    • Like in Example 3, the Transfer Function is determined through analysis of the entire system.

    Example 5: Two Mass System (No Friction)

    • Two masses m1 and m2 are connected by springs K1, K2, K3.
    • There is no friction in this example.
    • An external force u(t) is applied to the first mass m1.
    • The output is displacement y(t) of the second mass m2.
    • The Transfer Function is simplified due to the absence of friction.

    Example 6: Two Mass System (Simplified)

    • This system has two masses m1 and m2 linked by springs and dampers.
    • It is similar to previous examples, but with different input and output points.
    • The Transfer Function is calculated through a similar process of force balance and Laplace transform application.

    Example 7: Two Mass System with Two Inputs

    • This system has two masses m1 and m2 with distinct forces u1(t) and u2(t) applied.
    • The output is the displacement of the second mass m2 (y2(t)).
    • The Transfer Function becomes more complex as it must accommodate two independent inputs.

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    Description

    This quiz covers the fundamental concepts of mechanical systems involving springs, dampers, and transitional systems. It includes the derivation of the transfer function and applications of Newton's Second Law. Test your understanding of these principles and their application in single mass systems.

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