ME 416 - Mechanical Vibrations

Institution:

Point Park University

Subject:

Description:

The course begins with consideration of a simple, unforced, helical spring-mass system. Free body diagrams (FBD) for the unloaded, static and dynamics conditions are used to produce an equation for the net force acting on the mass. This force is separately determined via inertial analysis. Together the FBD and inertial relationships form the differential equation of motion. The "D" operator method is used to produce the solution in terms of imaginary exponentials and the Euler equations are used to convert the solution to one in terms of Sines and Cosines. Initial values of displacement and velocity are used to determine coefficients which stem from the constants of integration. With minor variations the above process towards a solution is followed in more complicated situations involving damping, forcing and multiple degrees of freedom. Rotational vibrations of torsion bars and leaf springs are analyzed. A short exercise in fluid mechanics is used to show that mechanical energy extraction by a hydraulic damper is dependent upon mass velocity. Solutions to unforced arrangements involving springs and dampers with a single mass are solved using the equivalent system and torsion analysis approaches. When a spring mass damper system is subjected to continuous forcing the differential equation of motion is seen to have a complementary function part which involves system characteristics and a particular integral part which involves forcing function form. The solution is seen to have a part which decays with time and a steady state part. The latter part is emphasized and the method of undetermined coefficients is used as a means of solution. The phenomena of beats and resonance are examined. The Duhamel integral is used in solutions when forcing exists over an initial finite interval. Matrix methods are applied to solve the coupled set of equations of motion resulting from unforced multi-mass systems. The course closes with the examination of situations involving both linear and rotational coordinates. Prerequisites: MATH 230, MATH 310. Course Objectives Upon successful completion of the course, students will be able to: (1) Find the natural frequency and periodicity of systems having a single mass attached to a helical spring, a torsion bar or a leaf spring. (2) Find the natural frequency and logarithmic decrement of amplitude of a single mass system which is either directly or indirectly connected to springs and hydraulic dampers. (3) Use either the equivalent system or the torsional analysis approach to a solution. (4) Solve problems involving continuous forcing of the spring mass damper system. (5) Know the circumstances under which resonance and beating will occur. (6) Calculate the time dependent response to a prolonged (as opposed to instantaneous) disturbance. (7) Find the principle frequency and modes of oscillating multi-mass systems. (8) Solve problems involving the use of linear and rotational coordinates.

Credits:

3.00

Credit Hours:

Prerequisites:

Corequisites:

Exclusions:

Level:

Instructional Type:

Lecture

Notes:

Additional Information:

Historical Version(s):

Institution Website:

www.pointpark.edu

Phone Number:

(412) 391-4100

Regional Accreditation:

Middle States Association of Colleges and Schools

Calendar System:

Semester

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