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Questions and Answers
What is the correct expression for the transfer function from u(s) to x1(s)?
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?
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?
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?
What represents the forces acting on mass m1 according to Newton's second law?
Which statement is true regarding the simplification of the equation for mass m2?
Which statement is true regarding the simplification of the equation for mass m2?
What does the variable K represent in the spring force equation F(t) = K x(t)?
What does the variable K represent in the spring force equation F(t) = K x(t)?
In the Laplace domain, how is the force generated by friction defined?
In the Laplace domain, how is the force generated by friction defined?
Which equation represents the transfer function (T.F.) in a transitional system?
Which equation represents the transfer function (T.F.) in a transitional system?
What is the result of applying Newton's Second Law in a mass-spring-damper system?
What is the result of applying Newton's Second Law in a mass-spring-damper system?
How is the input force represented in the force equation for mass m1 of a two-mass system?
How is the input force represented in the force equation for mass m1 of a two-mass system?
What does the equation F(s) = x(s) [ms² + BS + K] imply about the relationship between force and displacement in the Laplace domain?
What does the equation F(s) = x(s) [ms² + BS + K] imply about the relationship between force and displacement in the Laplace domain?
When analyzing a damped system, what role does the damping constant B play in the force equation F(t) = Bx(t)?
When analyzing a damped system, what role does the damping constant B play in the force equation F(t) = Bx(t)?
What is the purpose of a free body diagram in the context of mechanical systems?
What is the purpose of a free body diagram in the context of mechanical systems?
What is the general form of the transfer function from x(s) to y(s) as derived from the system?
What is the general form of the transfer function from x(s) to y(s) as derived from the system?
Which equation correctly represents the forces acting on mass m2?
Which equation correctly represents the forces acting on mass m2?
What expression describes the final transfer function T.F for the system?
What expression describes the final transfer function T.F for the system?
In the context of the Laplace Transform, what does the variable s commonly represent?
In the context of the Laplace Transform, what does the variable s commonly represent?
From the equation K2y(s) = z(s) [m2s² + B2s + K3 + K2], what does z(s) represent?
From the equation K2y(s) = z(s) [m2s² + B2s + K3 + K2], what does z(s) represent?
Which relationship correctly expresses the connection between input force F(s) and output x(s) for mass m1?
Which relationship correctly expresses the connection between input force F(s) and output x(s) for mass m1?
What is the coefficient of the damping factor within the equation for mass m2?
What is the coefficient of the damping factor within the equation for mass m2?
Which of the following is NOT a component of the transfer function from y(s) to z(s)?
Which of the following is NOT a component of the transfer function from y(s) to z(s)?
What is the form of the transfer function from x(s) to y(s)?
What is the form of the transfer function from x(s) to y(s)?
Which equation represents the forces acting on mass m2?
Which equation represents the forces acting on mass m2?
What is the final form of the transfer function (T.F.) given in the content?
What is the final form of the transfer function (T.F.) given in the content?
In the system described, what happens to the transfer function from u1(s) to y2(s) when u2(s) is set to 0?
In the system described, what happens to the transfer function from u1(s) to y2(s) when u2(s) is set to 0?
What is the equation for finding the relationship between x(s) and y(s) represented in terms of the given parameters?
What is the equation for finding the relationship between x(s) and y(s) represented in terms of the given parameters?
What is the purpose of applying the Laplace transform in the analysis of the system?
What is the purpose of applying the Laplace transform in the analysis of the system?
What values do G1, G2, and G3 represent in the context of the transfer function calculations?
What values do G1, G2, and G3 represent in the context of the transfer function calculations?
How is the final transfer function derived from G1, G2, and G3?
How is the final transfer function derived from G1, G2, and G3?
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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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