{"id":1630,"date":"2025-11-23T13:29:35","date_gmt":"2025-11-23T14:29:35","guid":{"rendered":"https:\/\/r-maher.com\/?p=1630"},"modified":"2025-11-23T13:44:27","modified_gmt":"2025-11-23T14:44:27","slug":"modal-and-harmonic-analysis-of-a-langevin-type-ultrasonic-transducer-using-full-3d-finite-element-modeling","status":"publish","type":"post","link":"https:\/\/r-maher.com\/?p=1630","title":{"rendered":"Modal and Harmonic Analysis of a Langevin-Type Ultrasonic Transducer Using Full 3D Finite Element Modeling"},"content":{"rendered":"

Ultrasonic transducers play a critical role in applications such as structural health monitoring, industrial cleaning, welding, medical imaging, and distance measurement.
Among various designs, the Langevin-type sandwich transducer<\/strong> is widely recognized for its strong electromechanical efficiency and robust resonance behavior.<\/p>\n

In this study, a complete three-dimensional finite element model<\/strong> of a Langevin transducer was developed. The objectives were to extract its natural frequencies, evaluate harmonic response under electrical excitation, and characterize the dynamic behavior for future tuning and optimization.<\/p>\n

\n

\"\"<\/strong><\/h3>\n

<\/h3>\n

Key Factors Analyzed during FE Simulations\"\"<\/strong><\/h3>\n
    \n
  • \n

    Identification of bending, mixed bending\u2013compression, axial, and rotational modes<\/p>\n<\/li>\n

  • \n

    Full electromechanical coupling of the piezoelectric stack<\/p>\n<\/li>\n

  • \n

    Harmonic response evaluation using the full 3D model<\/strong><\/p>\n<\/li>\n

  • \n

    Correlation between modal frequencies and harmonic amplitude peaks<\/p>\n<\/li>\n

  • \n

    Effects of clamped installation on mode shapes<\/p>\n<\/li>\n

  • \n

    Visualization of bending and rotational behavior across lateral directions<\/p>\n<\/li>\n<\/ul>\n

    \n

    \"\"<\/strong><\/h3>\n

    <\/h3>\n

    Project Highlights<\/strong><\/h3>\n
      \n
    • \n

      Geometry:<\/strong> Full 3D Langevin-type structure<\/p>\n<\/li>\n

    • \n

      Material:<\/strong> Complete piezoelectric electromechanical definition<\/p>\n<\/li>\n

    • \n

      Analysis Types:<\/strong> Modal + Harmonic<\/p>\n<\/li>\n

    • \n

      Boundary Condition:<\/strong> Fully fixed at the top surface<\/p>\n<\/li>\n

    • \n

      Excitation:<\/strong> Electrical voltage applied on electrode faces<\/p>\n<\/li>\n

    • \n

      Modeling Approach:<\/strong> Full geometry used for both modal and harmonic analysis<\/p>\n<\/li>\n

    • \n

      Goal:<\/strong> Dynamic characterization, validation, and performance assessment<\/p>\n<\/li>\n<\/ul>\n

      \n

      \"\"<\/strong><\/h3>\n

      <\/h3>\n

      FE Analysis Tips and Tricks<\/strong><\/h3>\n
        \n
      • \n

        Use coupled-field solid elements to ensure correct electromechanical behavior.<\/p>\n<\/li>\n

      • \n

        Apply bonded contacts throughout the piezo\u2013metal interfaces for realistic stress transfer.<\/p>\n<\/li>\n

      • \n

        Refine mesh in the piezoelectric region; ultrasonic systems are highly mesh-sensitive.<\/p>\n<\/li>\n

      • \n

        Always perform harmonic analysis on the full model<\/strong> to maintain correct stiffness path and dynamic response.<\/p>\n<\/li>\n

      • \n

        Clamped boundary conditions significantly modify the modal landscape \u2014 ensure they reflect actual mounting conditions.<\/p>\n<\/li>\n

      • \n

        A fine frequency resolution near expected peaks improves harmonic accuracy.<\/p>\n<\/li>\n<\/ul>\n

        \n

        Material Selection<\/strong><\/h3>\n

        The piezoelectric disk stack was assigned a complete set of electromechanical properties including:<\/p>\n

          \n
        • \n

          Elastic stiffness matrix<\/p>\n<\/li>\n

        • \n

          Piezoelectric coupling matrix<\/p>\n<\/li>\n

        • \n

          Relative permittivity tensor<\/p>\n<\/li>\n

        • \n

          Density and damping<\/p>\n<\/li>\n<\/ul>\n

          Metallic front and back masses were modeled using standard linear elastic properties.<\/p>\n

          \n

          Geometry and Mesh Generation<\/strong><\/h3>\n

          The transducer model included:<\/p>\n

            \n
          • \n

            Front mass<\/p>\n<\/li>\n

          • \n

            Back mass<\/p>\n<\/li>\n

          • \n

            Piezoelectric stack<\/p>\n<\/li>\n

          • \n

            Central bolt<\/p>\n<\/li>\n

          • \n

            Electrode layers<\/p>\n<\/li>\n<\/ul>\n

            A tetrahedral mesh with local refinement ensured high accuracy in deformation and wave propagation prediction.<\/p>\n

            \"\"<\/strong><\/h3>\n
            \n

            Boundary Conditions<\/strong><\/h3>\n
              \n
            • \n

              Full fixation on the top surface<\/strong>, representing the real clamped installation<\/p>\n<\/li>\n

            • \n

              Electrical potential applied to piezoelectric disks during harmonic response<\/p>\n<\/li>\n

            • \n

              Bonded mechanical interfaces across all assembly components<\/p>\n<\/li>\n<\/ul>\n

              \n

              Analysis Settings<\/strong><\/h3>\n

              Modal Analysis<\/strong><\/p>\n

                \n
              • \n

                Extraction of natural frequencies up to ~6 kHz<\/p>\n<\/li>\n

              • \n

                Identification of bending, axial, and rotational modes<\/p>\n<\/li>\n<\/ul>\n

                Harmonic Analysis<\/strong><\/p>\n

                  \n
                • \n

                  Full transducer model used<\/p>\n<\/li>\n

                • \n

                  Voltage-driven excitation<\/p>\n<\/li>\n

                • \n

                  Frequency sweep across the same range<\/p>\n<\/li>\n

                • \n

                  Displacement amplitudes captured in both lateral and longitudinal directions<\/p>\n<\/li>\n<\/ul>\n

                  \n

                  Results Interpretation<\/strong><\/h2>\n

                  Modal Results<\/strong><\/h3>\n

                  The first structural modes appeared at the following frequencies:<\/p>\n

                    \n
                  • \n

                    1153.5 Hz<\/strong><\/p>\n<\/li>\n

                  • \n

                    1154.3 Hz<\/strong><\/p>\n<\/li>\n

                  • \n

                    2447 Hz<\/strong><\/p>\n<\/li>\n

                  • \n

                    2457.3 Hz<\/strong><\/p>\n<\/li>\n

                  • \n

                    4762 Hz<\/strong><\/p>\n<\/li>\n

                  • \n

                    5448.2 Hz<\/strong><\/p>\n

                    \"\"<\/strong><\/h3>\n<\/li>\n<\/ul>\n

                    These frequencies correspond to:<\/p>\n

                      \n
                    • \n

                      Bending-dominated modes in the lateral directions<\/p>\n<\/li>\n

                    • \n

                      Mixed bending\u2013compression shapes<\/p>\n<\/li>\n

                    • \n

                      Higher-order bending patterns<\/p>\n<\/li>\n

                    • \n

                      Rotational modes around the longitudinal axis<\/strong>, which naturally appear in clamped transducer structures<\/p>\n<\/li>\n<\/ul>\n

                      \n

                      Harmonic Response Results<\/strong><\/h3>\n

                      Using the full 3D model<\/strong>, the harmonic frequency sweep revealed displacement peaks at approximately:\"\"<\/p>\n

                        \n
                      • \n

                        ~1200 Hz<\/strong><\/p>\n<\/li>\n

                      • \n

                        ~2400\u20132500 Hz<\/strong><\/p>\n<\/li>\n

                      • \n

                        ~5400\u20135500 Hz<\/strong><\/p>\n<\/li>\n<\/ul>\n

                        These peaks align with the modal frequencies and reflect lateral bending modes, higher-order bending effects, and slight rotational coupling around the transducer\u2019s longitudinal axis.<\/p>\n

                        The longitudinal deformation remained limited due to the clamped boundary condition, which suppresses high-frequency axial resonance.<\/p>\n

                        \n

                        Engineering Insights<\/strong><\/h3>\n
                          \n
                        • \n

                          When the transducer is clamped, bending modes dominate the dynamic behavior, and classic longitudinal ultrasonic resonance does not appear in the operating range.<\/p>\n<\/li>\n

                        • \n

                          Harmonic amplitude peaks matching modal frequencies confirm correct material definition and model fidelity.<\/p>\n<\/li>\n

                        • \n

                          Rotational modes around the main axis emerge naturally due to asymmetry in clamped conditions and should be considered during design tuning.<\/p>\n<\/li>\n

                        • \n

                          Using the full model<\/strong> avoids artificial symmetry constraints and provides realistic predictions of both bending and rotational behavior.<\/p>\n<\/li>\n

                        • \n

                          This simulation workflow is suitable for optimizing resonance performance, improving mechanical robustness, and designing tailored frequency responses.<\/p>\n<\/li>\n<\/ul>\n

                          \n

                          Conclusions<\/strong><\/h3>\n
                            \n
                          1. A complete 3D electromechanical model of the Langevin-type transducer was successfully developed.<\/li>\n
                          2. Modal analysis revealed bending, mixed bending, and rotational modes in the 1\u20136 kHz range.<\/li>\n
                          3. Harmonic analysis using the full model<\/strong> produced amplitude peaks aligned with modal frequencies, confirming model accuracy.<\/li>\n
                          4. The clamped installation condition significantly shifts dynamic behavior toward lateral and rotational modes.<\/li>\n
                          5. This validated modeling framework provides a reliable foundation for future transducer optimization and resonance tuning.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

                            Ultrasonic transducers play a critical role in applications such as structural health monitoring, industrial cleaning, welding, medical imaging, and distance…<\/p>\n","protected":false},"author":2,"featured_media":1633,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[157,152,150,154,149,155,148,90,158,151,153,147,156],"class_list":["post-1630","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-analysis","tag-dynamic-characterization","tag-electromechanical-simulation","tag-finite-element-analysis","tag-frequency-response","tag-harmonic-response","tag-hyperelastic-coupled-field-simulation","tag-langevin-transducer","tag-modal-analysis","tag-piezoelectric-material","tag-piezoelectric-modeling","tag-structural-dynamics","tag-ultrasonic-transducer","tag-vibration-analysis"],"acf":[],"_links":{"self":[{"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/posts\/1630","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/r-maher.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1630"}],"version-history":[{"count":6,"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/posts\/1630\/revisions"}],"predecessor-version":[{"id":1643,"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/posts\/1630\/revisions\/1643"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/r-maher.com\/index.php?rest_route=\/wp\/v2\/media\/1633"}],"wp:attachment":[{"href":"https:\/\/r-maher.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1630"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/r-maher.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1630"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/r-maher.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1630"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}