Physical & Mathematical Modeling Problems: Mechanical Systems with Solutions. K. Craig 1

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Physical & Mathematical Modeling Problems:

Mechanical Systems

Solutions with

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# 1

A high-speed solenoid actuator-valve system is shown. This type of actuator is used in hydraulic systems to position spool valves for metering fluid flow.

Electric current flows through the coil of wire that surrounds the plunger and generates a magnetic field, which produces an attracting force on the plunger, pulling it to the right. Hence, the electromagnetic force pulls the plunger

toward the center of the coil and closes the air gap. The plunger is connected to the spool valve via the push rod. The return spring is used to return the

plunger-valve mass back to its seated position when the electromagnetic force is removed.

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# 2

Shown is a vibration-isolation system that improves the ride quality for a commercial vehicle, such as a tractor-trailer used for long-distance transportation of freight. Travel over a rough road causes vibrations that are transmitted to the vehicle’s cabin floor, and the floor vibrations, z₀(t), are transmitted to the seat mass m₁ and ultimately to the driver (mass m₂). A properly designed seat-suspension system will suppress the road vibrations transmitted to the driver. The seat suspension consists of a passive shock absorber,

modeled by pure/ideal spring and damper elements (k₁ and b₁). The damping and stiffness of the seat cushion are modeled by pure/ideal spring and damper elements (k₂ and b₂).

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# 3

Shown is a piezoelectric actuator for manufacturing MEMS where high-accuracy positioning is required. It is designed to maintain contact with a slide mass and move it to a desired position. The actuator uses two sets of ceramic piezoelectric materials (e.g., lead zirconate titanate or PZT exhibits a mechanical deformation when a voltage is applied) to provide

external forces on the clamp mass m₁. A vertical stack of PZT layers (not shown) provides a vertical force that clamps mass m₁ to the slide mass m₂. The horizontal stack of PZT layers extends when a voltage is applied and, therefore, pushes mass m₁ to the right. The slide mass m₂ can then be moved by friction to a desired horizontal location by a “clamp-extend- release” sequence.

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# 4

Shown is a wind turbine generator used for transforming mechanical energy into electrical energy. The turbine inertia J₁ consists of the wind turbine blades, turbine shaft, and gear 1 and the generator inertia J₂ consists of gear 2,

generator shaft, and the generator rotor. This assumes that the shafts are rigid, and not compliant. Both the turbine and generator inertias experience viscous

friction. The turbine blades extract energy from the wind and produce the

aerodynamic torque T_aero, which is the input to the system. The generator disk J₂ includes a coil of wire, and the rotational motion of the wire windings in a magnetic field generates electrical energy. In addition, the wire coil carrying current in the magnetic field experiences an induced force that opposes the motion of the generator (represented by T_gen).

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# 5

The figure shows a dual-disk mechanical system that has been proposed as an efficient generator for hybrid vehicles. The mechanical system is composed of a toroidal-segment piston (disk J₁) matched with a toroidal- segment cylinder (disk J₂). Both disks rotate about a common axis. The disks are connected by a torsional spring. Angular displacements ϴ₁

(piston disk) and ϴ₂ (cylinder disk) are measured from their equilibrium positions. Both disks experience friction, modeled as viscous friction. A gas-pressure torque from a diesel engine, T_in(t), drives the two-disk

system in equal-and-opposite pairs. Angular rotation of the disks relative to stationary magnets generates electrical current and reaction torques, but these effects are not included here. During the normal operating mode, the input engine torque T_in(t) is pulsed so that the elastic system deflects in a manner such that both disks vibrate about a mean angular motion in one direction.

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# 6 Assorted Basic Mechanical Modeling Problems

(a) (b)

(c) (d)

(e)

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# 7 Assorted Basic Mechanical Modeling Problems

(a)

(b)

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# 8

Assorted Basic Mechanical Modeling Problems

(a)

(b)

(c)

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# 9 Assorted Basic Mechanical Modeling Problems (a)

(b)

(c)

(d)

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# 10

A fluid coupling system is shown. It consists of the impeller disk J₁, turbine disk J₂, and load disk J₃. Each disk has an independent angular displacement variable, ϴ₁, ϴ₂, and ϴ₃, which are all measured from the equilibrium (zero twist) position. An external torque T_in(t) is applied directly to the impeller disk J₁, which transmits torque to the turbine disk J₂ because of their relative angular velocity and the viscous

friction of the hydraulic fluid. J₁ and J₂ are not mechanically

connected. The impeller and turbine disks are contained in a sealed housing filled with hydraulic fluid. A flexible shaft connects the turbine disk J₂ to the load disk J₃. An external load torque T_L acts directly on disk J₃.

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# 11

A gear train driving a robot arm through a flexible shaft is shown. An external motor (not shown) delivers the input torque T_in to the input gear 1. The moment of inertia of disk 1 and gear 2 is J₁. Gear 2 is much larger than gear 1. The gear train and robot arm are connected by a flexible shaft. The robot arm has inertia J_cm about its center of mass and its center of mass is distance d from the rotation axis. The weight of the robot arm provides an opposing torque on the arm about the rotation axis for angular displacement 0 < ϴ₂ < 180°. The system has negligible friction.

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# 12

Shown is an optical disk drive system. The spindle motor (contained in the chassis) rotates the disk and the cart servo motor translates the pick-up head along the radial direction of the spinning disk so that the focused laser reads the desired data track on the optical disk.

Although the cart servo motor can position the cart in the radial

direction, the cart is rigidly attached to the chassis and so together are a lumped mass. A series of rubber mounts connects the chassis to the frame. The mounts are used to suppress transmitted vibrations

from the motion of the frame. The absolute displacement of the frame is x_in(t).

m₁

m₂

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# 13

High-speed electric trains use a mechanical arm called a pantograph to transfer electric current from an overhead wire to the train. The pantograph typically consists of a two-arm frame linkage that provides an upward force in order to maintain contact between a small pan-head and the catenary wire. A two-lumped mechanical physical model of the pantograph is shown. m₁ is the head mass, m₂ is the frame mass, and k₁ is the stiffness of the shoe contact between the head and the catenary wire (only compression possible as the wire cannot pull on the head mass). The head suspension is modeled as a pure/ideal spring/damper, while the frame suspension is modeled as a pure/ideal damper. A pneumatic piston

provides the force f_a(t) that pushes up on the frame so that the shoe remains in contact with the wire.

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# 14

A schematic of an automotive valve train is shown. The rotating cam moves the pushrod, and the rocker arm rotates to move the valve in the vertical direction. A simplified mechanical model is shown. Moment if inertia J represents the rocker-arm inertia about the pivot point. Deflection of the pushrod is modeled by the spring k₁, and the return spring on the valve is k₂. Friction in the rocker-arm pivot is modeled by viscous friction coefficient b. The vertical motion of the cam follower is x_C(t), the input to the system. When the rocker-arm angle is level (ϴ = 0), the return spring has a compressive preload force of F_L. When cam follower position x_C = 0 and ϴ = 0, then the pushrod is undeflected. Assume that the rocker arm angle ϴ remains small at all times.

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# 15

The figure shows a model of a locomotive pulling two railroad cars.

The two couplers are modeled by pure/ideal springs and dampers.

Assume that the rolling friction of each mass is equal to the product of friction coefficient b_r and absolute velocity dz_i/dt. The

locomotive’s propulsion system provides external force F_a(t) to mass m₁.

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# 16

The figure shows a ¼-car model that is used to analyze the ride quality of automotive suspension systems. Mass m₁ is the “sprung mass, which is one-quarter of the vehicle mass that is supported by the suspension system. Mass m₂ is the “unsprung mass,” which is the lumped mass

composed of one wheel and half-axle assembly, plus the shock absorber and suspension springs. Tire stiffness is modeled by spring k₂. The

suspension system is modeled by a pure/ideal spring and damper. The input is road position, z_in(t), which is measured with respect to a level road datum.

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# 17

The figure shows a lumped-parameter representation of a concept for a MEMS “tuning fork gyroscope” for measuring angular velocity.

Two external forces f₁ and f₂ are applied to masses m₁ and m₂, respectively.

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