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Solving Engineering Problems in Dynamics

Solving Engineering Problems in Dynamics

Michael Spektor


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“Dr. Spektor’s new and independent scholarship on the use of the Laplace Transform is profound.” 

—Professor Lawrence J. Wolf, Oregon Institute of Technology

To read the full review, click on the Resources tab above.



Solving Engineering Problems in Dynamics helps practicing engineers successfully analyze real mechanical systems by presenting comprehensive methods for analyzing the motion of engineering systems and their components. This analysis covers three basic phases:  1) composing the differential equation of motion; 2) solving the differential equation of motion; and 3) analyzing the solution.


Although a formal engineering education provides the fundamental skills for completing these phases, many engineers nonetheless would benefit by gaining further insight in using these fundamentals to solve real-life engineering problems. This book thus describes in step-by-step order the methods related to each of these phases.



  • A basic education in engineering is sufficient to master the contents of this guide and effectively apply its step-by-step methods for solving engineering problems.
  • Numerous solutions of examples of linear, non-linear, and two-degree-of-freedom systems are found throughout.
  • Explains the structures of differential equations of motion of the two-degree-of-freedom systems and demonstrates the applicability of the Laplace Transform methodology for solving these equations.
  • Many types of engineers can benefit from this book (as well as students in mechanical, manufacturing, and industrial engineering).

Michael B. Spektor taught for many years at Oregon Institute of Technology, and before retiring he was the director of the manufacturing engineering technology bachelor degree program at Boeing in Seattle. He has an undergraduate degree in mechanical engineering from Kiev Polytechnic University and a Ph. D. in mechanical engineering from Kiev Construction University. He has worked in both industry and higher education in the United States, Israel, and the former Soviet Union. Spektor holds five U.S. Patents and two U.S.S.R. Inventor’s Certificates. Some of his career highlights include: chief designer of an automobile crane; the design and development of vibratory and impact machines; an analysis of the dynamics of construction safety harnesses that directly led to their improvement; developer of the theory and engineering calculations for the optimization of soil-working vibratory processes for minimum energy consumption; analytical investigations of media deformation under dynamic loading that improved the methodologies for measuring and interpreting experimental data; and the publication of numerous scientific articles on dynamics.  

Review of Solving Engineering Problems in Dynamics

"In 54 years as a mechanical engineer and scholar, I have never had Laplace Transforms laid out for me in a more complete and understandable manner than it is by Michael Spektor in his two-volume set of books Solving Engineering Problems in Dynamics and Applied Dynamics in Engineering. Now, I have to confess to a “secret.”

     Way back in the final year of my doctoral studies, I was advised that I ought to have a graduate-level math course listed among my studies. I found a course that was scheduled to be taught the next semester entitled something like “Transform Calculus.” I had A grades in calculus, and I had been a “whiz kid” with the Fourier series, so I signed up for it. 

     As you can imagine, there was a lot on my mind that semester. I found the course to be abstract and diverting. My disinterest was duly rewarded with a C grade—the only C that I had received since my freshman year. Since it was the only course that I took that term, I was surprised to receive a letter advising me that I was on academic probation and would have to take another graduate course the next semester and get an A grade to average out that C. I chose another highly technical course and got the A.

     Though professionally, I subsequently used other transform methods both analytical and experimental, the incident left me terrified of the Laplace Transform.  My fear continued to stalk me, even though shortly afterward in my career I successfully held the title of “Structural Dynamics Engineer” with a Fortune 500 company.

     So I was again surprised when Dr. Michael Spektor, my long-time friend and colleague for 26 years, told me that over his 10 years of retirement, he had just completed the two above-cited books devoted to the use of Laplace Transforms in the solution of mechanical engineering and technology problems. I knew Michael to be a very accurate and successful professor and department head. And I was familiar with his research work on designing a vibration machine to penetrate soil. But Dr. Spektor’s new and independent scholarship on the use of the Laplace Transform is profound. 

      He has searched the literature on transforms that would be specific to the study and practice of mechanical engineering only.  And he reduces his findings to 96 transform pairs that meet the specific needs of mechanical engineers. This I learned from him as I now enter the final year before my own retirement from teaching. I expect that checking through some of his many transform pairs will be an early pleasure of my own retirement and my own overdue conquest of this, my personal Chimera."


—Professor Lawrence J. Wolf, Oregon Institute of Technology




Differential Equations Of Motion

  • Analysis Of Forces
  • Analysis of Resisting Forces
  • Forces of Inertia
  • Damping Forces
  • Stiffness Forces
  • Constant Resisting Forces
  • Friction Forces
  • Analysis of Active Forces
  • Constant Active Forces
  • Sinusoidal Active Forces
  • Active Forces Depending on Time
  • Active Forces Depending on Velocity
  • Active Forces Depending on Displacement

Solving Differential Equations of Motion Using Laplace Transforms

  • Laplace Transform Pairs For Differential Equations of Motion
  • Decomposition of Proper Rational Fractions
  • Examples of Decomposition of Fractions
  • Examples of Solving Differential Equations of Motion
  • Motion by by Inertia with no Resistance
  • Motion by Inertia with Resistance of Friction
  • Motion by Inertia with Damping Resistance
  • Free Vibrations
  • Motion Caused by Impact
  • Motion of a Damped System Subjected to a Tim Depending Force
  • Forced Motion with Damping and Stiffness
  • Forced Vibrations

Analysis of Typical Mechanical Engineering Systems

  • Lifting a Load
  • Acceleration
  • Braking
  • Water Vessel Dynamics
  • Dynamics of an Automobile
  • Acceleration
  • Braking
  • Acceleration of a Projectile in the Barrel
  • Reciprocation Cycle of a Spring-loaded Sliding Link
  • Forward Stroke Due to a Constant Force
  • Forward Stroke Due to Initial Velocity
  • Backward Stroke
  • Pneumatically Operated Soil Penetrating Machine

Piece-Wise Linear Approximation

  • Penetrating into an Elasto-Plastic Medium
  • First Interval
  • Second Interval
  • Third Interval
  • Fourth Interval
  • Non-linear Damping Resistance
  • First Interval
  • Second Interval

Dynamics of Two-Degree-of-Freedom Systems

  • Differential Equations of Motion: A Two-Degree-of-Freedom System
  • A System with a Hydraulic Link (Dashpot)
  • A System with an Elastic Link (Spring)
  • A System with a Combination of a Hydraulic Link (Dashpot) and an Elastic Link (Spring)
  • Solutions of Differential Equations of Motion for Two-Degree-of-Freedom Systems
  • Solutions for a System with a Hydraulic Link
  • Solutions for a System with an Elastic Link
  • Solutions for a System with a Combination of a Hydraulic and an Elastic Link
  • A System with a Hydraulic Link where the First Mass Is Subjected to a Constant External Force
  • A Vibratory System Subjected to an External Sinusoidal Force