Problems
Problem 2.1 (PLAYMATE)
Consider the illustration of figure 2.7 in which a bending plate scale is to have a heavy load placed upon it. Such scales measure the weight of the load by measuring the strain on the sensors and electronically converting this to the weight placed on the plate. (It goes without saying that calibration is required for such systems.)
It takes time for the system to come to equilibrium, during which oscillation occurs. Develop a one-dimensional lumped-parameter model of the mechanical aspect of the system and its applied load, via the following steps.
Declare what you will take to be the system and its input(s).
Declare a one-dimensional, mechanical, lumped-parameter model for the system. How might you determine the lumped-parameter model parameters (e.g. mass, spring constant, etc.)?
Draw a schematic of the lumped-parameter system model.
Draw a linear graph corresponding to your lumped-parameter model.
There are two good approaches here. The first approach is to take the scale as the system and the load as a force input. The second approach is to take the load as part of the system once it’s in contact with the scale and take no inputs, instead modeling the transient response as an initial velocity condition. Either approach is valid, and both will be developed in parallel in this solution. A student could have used either approach and needn’t have developed both.
In both approaches, we take the vertical direction as our one-dimension. In both approaches, we will use a single spring-mass-damper lumped-parameter model. In both approaches, the spring-like behavior of the steel plate will be modeled as a mechanical spring \(k\). In both approaches, the energy losses to heat will be modeled as a linear damper \(B\). In the first approach, the mass of the plate \(m_1\) will be taken as the lumped mass. In the second approach, the combined mass \(m_2\) of the plate and load will be taken as the lumped mass.
Determining the parameters \(k\), \(B\), \(m_1\), and \(m_2\) can proceed by both analysis and measurement. For \(k\), analysis of a rectangular cross-sectional beam’s load/displacement could yield an analytic estimate; measurement of displacement for different loads would also work. For \(B\), it is probably best to measure several responses and relate the motion decay rate and \(B\). For \(m_1\) and \(m_2\), density-geometry relations yield the mass or direct measurements of mass is reasonable.
Problem 2.2 (NOD)
Consider the illustration of figure 2.8 in which a motor is on a machine and an instrument is atop a nearby workbench. The motor typically spins at a fixed velocity, generating a vibration that is transmitted through the machine and into the floor.
Suppose you are given the task of designing the feet of the instrument such that less than a certain amount of vibratory motion from the motor will be transmitted through the floor and workbench to the instrument.
Develop a one-dimensional lumped-parameter model of the mechanical aspect of the system via the following steps.
Declare what you will take to be the system and its input(s).
Declare a one-dimensional, mechanical, translational, lumped-parameter model for the system.
Draw a schematic of the lumped-parameter system model.
Draw a linear graph corresponding to your lumped-parameter model.
There are at least two good approaches here. The first approach is to model the motor’s vibration as a force input, either on a mass representing the motor or on a mass representing the machine. The second approach is to model the motor’s vibration as a velocity input to the motor mount. Either approach is valid, and both will be developed in parallel in this solution. A student could have used either approach or another approach that is similarly valid.
For both approaches, the following statements hold.
We take the vertical direction as our one-dimension.
The machine is modeled as a mass mounted to the floor through a parallel spring-damper model.
The floor is modeled as a mass with a spring-damper model connecting to the building structure, which is assumed to be motionless.
The workbench and its legs are modeled as a mass with a parallel spring-damper model connecting it to the floor.
The instrument is modeled as a mass and its feet as a parallel spring-damper model connecting to the workbench.
In the first approach, the motor and its mount are modeled as a force source. In the second approach, the motor is modeled as a velocity source applied to the motor mount, modeled as a parallel spring-damper.
The following schematics shows the two good approaches described above.
Figure The linear graph for each approach is below. In both cases, the coordinate arrow is assigned downward.
Figure
Problem 2.3 (JOHNNYCASH)
Consider the illustration of figure 2.9 in which a wind turbine is harvesting wind energy. An electrical generator converts the rotational mechanical power into electrical energy.
Suppose you are given the task of designing the bearing and the flexible shaft coupler assemblies. For the design, you will need to know how wind speeds will affect the angular velocities and torques throughout the rotational mechanical system. Therefore, you resolve to develop a dynamic system model.
Develop a one-dimensional lumped-parameter model of the mechanical aspect of the system. Assume that the wind produces an input torque \(T_S\) via the turbine blades. Further assume that the generator draws power from the mechanical system in a manner that produces a load torque \(T_G\) that is proportional to the generator shaft angular velocity \(\Omega_G\); that is, for constant \(\beta\), \[\begin{aligned} T_G = \beta \Omega_G. \end{aligned}\]
Use the following steps:
Declare what you will take to be the system and its input(s).
Declare a one-dimensional, rotational mechanical, lumped-parameter model for the system.
Draw a schematic of the lumped-parameter system model.
Draw a linear graph corresponding to your lumped-parameter model.
Problem 2.4 (LILLIMOOMIE)
Finish applying the sign coordinate arrows on the following linear graphs.
electronic system
rotational mechanical system (assume \(T_S\) is in the positive direction)
translational mechanical system
Problem 2.5 (VARIETIES)
Draw necessary sign coordinate arrows and a linear graph for each of the following schematics.
electronic system, current source
rotational mechanical system, torque source
translational mechanical system, velocity source
The linear graph model is shown, below. Assign the coordinate arrows toward ground.
The linear graph model is shown, below. Assign the coordinate arrows toward the right.
The linear graph model is shown, below. Assign the coordinate arrows toward ground.
Problem 2.6 (CORMAC)
Draw necessary sign coordinate arrows and draw a linear graph for each of the following schematics.
electronic system, voltage source
rotational mechanical system, angular velocity source
translational mechanical system, force source
The linear graph model is shown, below.
The linear graph model is shown, below. Assign the coordinate arrows toward the right.
The linear graph model is shown, below. Assign the coordinate arrows toward ground.
Problem 2.7 (KURT)
Draw necessary sign coordinate arrows and a linear graph for each of the following schematics.
electronic system, voltage source
rotational mechanical system, torque source
translational mechanical system, force source
The linear graph model is shown, below.
The linear graph model is shown, below. Assign the coordinate arrows toward the right.
The linear graph model is shown, below. Assign the coordinate arrows toward ground.
Problem 2.8 (BUNKER)
Use the assigned coordinate arrows to draw a linear graph for each of the following schematics.
electronic system, voltage and current source
rotational mechanical system, torque source, coordinate arrow
translational mechanical system, force sources (2)
The linear graph model is shown, below.
The linear graph model is shown, below. Assign the coordinate arrows toward the right.
The linear graph model is shown, below. Assign the coordinate arrows toward ground.
Online Resources for Section 2.5
No online resources.