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Questions and Answers
What fundamental property must an impulse possess to induce a change in an object's motion?
What fundamental property must an impulse possess to induce a change in an object's motion?
- Any combination of force and time will result in the same change in momentum.
- Constant magnitude over an extended duration.
- Low magnitude and long duration.
- High magnitude and short duration. (correct)
How is impulse mathematically defined in relation to force and time?
How is impulse mathematically defined in relation to force and time?
- The average force divided by the duration of the impulse.
- The product of force and the square of the time interval.
- The integral of force over the time interval during which it acts. (correct)
- The derivative of force with respect to time.
In the context of dynamics, to what is the impulse imparted on an object directly equivalent?
In the context of dynamics, to what is the impulse imparted on an object directly equivalent?
- The object's final kinetic energy.
- The average force acting on the object.
- The change in the object's momentum. (correct)
- The object's potential energy.
Consider a stationary SDOF spring-mass-damper system subjected to an impulse. What is the immediate effect of the impulse on the system's velocity and displacement?
Consider a stationary SDOF spring-mass-damper system subjected to an impulse. What is the immediate effect of the impulse on the system's velocity and displacement?
What type of motion characterizes the impulse response of a system?
What type of motion characterizes the impulse response of a system?
For an under-damped system subjected to an impulse, what determines the specific form of the system's transient response?
For an under-damped system subjected to an impulse, what determines the specific form of the system's transient response?
If an impulse F occurs at time $t = \tau$, how is the transient response for $t < \tau$ characterized?
If an impulse F occurs at time $t = \tau$, how is the transient response for $t < \tau$ characterized?
What simplification is introduced when defining the unit impulse?
What simplification is introduced when defining the unit impulse?
If F represents the magnitude of an impulse, and $g(t - \tau)$ represents the response of the system to a unit impulse, how is the system's response $x(t - \tau)$ to the impulse F mathematically expressed?
If F represents the magnitude of an impulse, and $g(t - \tau)$ represents the response of the system to a unit impulse, how is the system's response $x(t - \tau)$ to the impulse F mathematically expressed?
What is the principal utility of convolution integral in the context of vibration analysis?
What is the principal utility of convolution integral in the context of vibration analysis?
In a mass-spring-damper system subjected to general loading, how is the response found using convolution integral?
In a mass-spring-damper system subjected to general loading, how is the response found using convolution integral?
Consider a system where $z = x - y$, with $z$ representing relative motion, $x$ absolute motion, and $y$ base excitation. How does this relative motion simplify the analysis?
Consider a system where $z = x - y$, with $z$ representing relative motion, $x$ absolute motion, and $y$ base excitation. How does this relative motion simplify the analysis?
When analyzing the response of a system subject to base excitation, what input is used to describe the force?
When analyzing the response of a system subject to base excitation, what input is used to describe the force?
If the transient response of a system to a unit impulse is $g(t - \tau)$, what is the system's response $x(t)$ to the force $F(\tau)$?
If the transient response of a system to a unit impulse is $g(t - \tau)$, what is the system's response $x(t)$ to the force $F(\tau)$?
In the context of using an impact hammer for vibration testing, what is the role of the load cell?
In the context of using an impact hammer for vibration testing, what is the role of the load cell?
What is the effect on the system’s response if a structure is subjected to two impacts, described by Dirac delta functions with magnitudes $F_1$ and $F_2$?
What is the effect on the system’s response if a structure is subjected to two impacts, described by Dirac delta functions with magnitudes $F_1$ and $F_2$?
How is the superposition principle applied when a system is subjected to multiple impulses at different times?
How is the superposition principle applied when a system is subjected to multiple impulses at different times?
What distinguishes the application of the convolution integral for an undamped system compared to a damped system?
What distinguishes the application of the convolution integral for an undamped system compared to a damped system?
When a mass-spring system is subjected to a constant force $F_o$ for $0 \le t \le t_o$, how does the response differ for $t \le t_o$ compared to $t > t_o$?
When a mass-spring system is subjected to a constant force $F_o$ for $0 \le t \le t_o$, how does the response differ for $t \le t_o$ compared to $t > t_o$?
What is the form of the generalized equation to determine the complete response of a system subjected to non-periodic loading?
What is the form of the generalized equation to determine the complete response of a system subjected to non-periodic loading?
How does increasing the mass of a single-degree-of-freedom system generally affect its response to an impulse load, assuming all other parameters remain constant?
How does increasing the mass of a single-degree-of-freedom system generally affect its response to an impulse load, assuming all other parameters remain constant?
For an overdamped system subjected to an impulse, how does its response differ from that of an underdamped system??
For an overdamped system subjected to an impulse, how does its response differ from that of an underdamped system??
In a situation where a machine is idealized as a spring-mass-damper system subjected to a step force, how does the damping ratio affect the system's settling time?
In a situation where a machine is idealized as a spring-mass-damper system subjected to a step force, how does the damping ratio affect the system's settling time?
Consider a SDOF system subjected to a force that is non-zero only over a finite time interval. After the force ceases, what type of vibration does the system exhibit?
Consider a SDOF system subjected to a force that is non-zero only over a finite time interval. After the force ceases, what type of vibration does the system exhibit?
Given a camcorder packed in a container with flexible packing material and dropped from a height h onto a rigid floor, what properties of the packing material influence the peak acceleration experienced by the camcorder?
Given a camcorder packed in a container with flexible packing material and dropped from a height h onto a rigid floor, what properties of the packing material influence the peak acceleration experienced by the camcorder?
In the context of a vehicle encountering a road bump, how does the vehicle's speed influence the amplitude and frequency of the resulting vibration?
In the context of a vehicle encountering a road bump, how does the vehicle's speed influence the amplitude and frequency of the resulting vibration?
When analyzing the vibration of a system subjected to nonperiodic loading, why are both superposition and convolution often necessary?
When analyzing the vibration of a system subjected to nonperiodic loading, why are both superposition and convolution often necessary?
Flashcards
Impulse
Impulse
A force applied for a very short time, large enough to cause a change in momentum.
Impulse Effect on Velocity
Impulse Effect on Velocity
The initial velocity change due to impulse while displacement remains unchanged.
Impulse Response
Impulse Response
Free vibration of a system, starting with an initial velocity after an impulse.
Transient response to impulse
Transient response to impulse
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Unit Impulse
Unit Impulse
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Response to any Impulse F at t=r
Response to any Impulse F at t=r
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General Force as Impulses
General Force as Impulses
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Convolution Integral
Convolution Integral
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Relative base excitation
Relative base excitation
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Study Notes
Impulse
- Impulse happens when a force is used for a very short time, but its influence is significant enough to change the momentum
- It is defined as the force integrated over the period it acts
Impulse Calculation
- F = ∫(t2,t1) F(t) dt = m(Xafter - Xbefore)
Impact on SDOF System
- The system is at rest (Xbefore = 0) at time t=0
- Velocity right after the impact in SDOF can be calculated using the equation Xafter = F/m
- Concept 1: Impulse changes the initial velocity (x) by F/m, but displacement value stays the same
- Concept 2: System's impulse response is like free vibration, with initial velocity being x = F/m
Transient Response to Impulse (Under-Damped)
- With initial mass condition (x = 0), initial condition depends on how the mass is damped
- To find the response to the impulse, a free-vibration expression can be used
- The initial velocity is: Xafter = Xo = F/m
- For underdamped mass (ζ < 1), use the free-response expression: x(t) = e^(-ζωnt) * (Xo*cos(ωdt) + ((Xo + ζωnXo)/ωd) * sin(ωdt))
- After inserting x=0 and x = F/m, the transient response will be, x(t) = (F*e^(-ζωnt) * sin(ωdt)) / (mωd)
Solution to Impulse at t=0
- Given the impulse happens at time t = 0
- For an impulse that occurs when t ≠0 a time (τ) is chosen when t > 0, rewritten as x(t) = 0 if 0<t<τ x(t) = (F*e^(-ζωn(t-τ)) * sin(ωd(t − τ))) / ωd if T<t
- The transient response is, x(t − ) = (F*e^(-ζωη (1-t)) sinωd(t − τ)) / mωd
- For an under-damped mass of 4kg, Wn = 10rad/s, ζ = 0.25, and an impulse of 8 N-s at τ = 0.2s, a different transient response appears
Unit Impulse
- It is possible to simplify solutions with generalization
- This is accomplished by defining the unit impulse as, F = ∫ Fdt = 1
- The response of an under-damped system to a unit impulse that happens at τ is g(t − ) = (e^(-ζωη(t-τ) * sin(ωd(t – t))) / mωd
- Using a previous expression for an under-damped system, the underdamped system's response to any impulse (F, at t = r) is the multiplication of impulse and system response to unit impulse, x(t − τ) = Fg(t − τ)
Transient Response of Systems
- Transient response systems can also be evaluated from free-response expressions for undamped and viscously damped systems
- Can find a response to a unit impulse, g(t − τ), for any value of damping and then find the response to any impulse again using, x(t − τ) = Fg(t − τ)
- The table provides response summary due to a unit impulse.
Response to General Force
- Convolution Integral
- The general force response, F(t), can be seen as seriers of magnitude impulses: F(ti)∆t
- When superposition is used: x(t) = x1(t) +x2(t) + x3(t) + ...
- Thus, x(t) = ΣF(τ)g(t − τ)∆τ
- Mass-spring-damper system response that has general loading can be found using convolution integral: x(t) = ∫(t,0) F(τ)g(t − τ)dτ
Base Excitation Response
- If z = x − y shows how mass moves relative to the base, the motion equation becomes, mx + c (x − ÿ) +k (x - y) = 0
- Can also be displayed as, m(ž + ÿ) + cž + kz = 0, or mž + cž + kz = -mÿ
- The response is explained with z(t) where F = -my: z(t) = (1/Wd) ∫(t,0) (t)e^(-ζωη(t-τ) sin(ωd(t − τ))dτ
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